1    PP 
1 


ill! 


1 

1 

l! 

111! 

lllliii 

iftll 

llilf iliilifliiliillliiiliilillii^i ' 

II  iifiij  iiji  in 

lillihliliiihllrlrif  ill 

$!ltill  iiHtsBBB^^H^H 

HBIllil  1  fill  Hll  !ll*ilt 

111 


mi  til    is  i 


frfo 


UNIVERSITY   OF   CALIFORNIA 


DEPARTMENT  OF  EDUCATION 


GIFT  OF  THE   PUBLISHER 


No. 


s 


Education  Department 


NEW 

GRAMMAR   SCHOOL 
ARITHMETIC 


BY 


JOHN   H.    WALSH 

ASSOCIATE  SUPERINTENDENT   OF  SCHOOLS,   THE  CITY 
OF  NEW  YORK 


BOSTON,  U.S.A. 
D.   C.   HEATH  &   CO.,   PUBLISHERS 
1905 


Copyright,  1895  and  1903, 
By  JOHN  H.  WALSH. 


INTRODUCTION. 


The  New  Grammar  School  Arithmetic  forms  with  the 
New  Primary  Arithmetic  a  complete  course  in  elementary 
school  mathematics. 

Each  of  the  first  four  chapters  of  the  New  Grammar 
School  Arithmetic  provides  for  a  half  year,  beginning  with 
advanced  matter,  which  is  followed  by  a  review  and  an 
extension  of  the  topics  of  the  preceding  grades.  Each  of 
the  next  two  chapters  (V  and  VI)  contains  arithmetic  work 
for  a  year,  which  should  be  supplemented  by  portions  of 
the  algebraic  and  geometrical  material  of  Chapters  VII  and 
VIII.  It  is  recommended  that  at  least  a  portion  of  the 
work  in  equations  of  Chapter  VII  should  precede  the  study 
of  Chapter  V. 

Among  the  special  features  of  the  New  Grammar  School 
Arithmetic  are  the  number  and  the  variety  of  the  problems ; 
the  systematic  reviews,  which  cover  oral  and  written  drill 
work  even  in  the  fundamental  operations ;  the  attention 
paid  to  short,  direct,  business  methods  of  computation ;  and 
the  spiral  handling  of  the  various  topics. 


iii 

548N48 


CONTENTS. 


CHAPTER  I. 

PAGES 

Mixed  Numbers 1  to  16 

Addition,  Subtraction,  Multiplication,  Division  (Common 

Denominators  determined  by  inspection). 
Review  of  Simple  Numbers 17  to  24 

Notation  and  Numeration,  Special  Drills,  Fundamental 

Processes. 
Decimals  (three  places) 25  to  32 

Addition  and  Subtraction,  Multiplication  and  Division 

by  an  Integer. 
United  States  Money  .         .        .         .        .         .         .         .  32  to  43 

Fractional  Parts  of  a  Dollar,  Division  of  United  States 

Money. 
Denominate  Numbers 43  to  45 

Time,  Dry  and  Liquid  Measures,  Avoirdupois  Weight, 

Miscellaneous  Examples. 
Measurements 46  to  49 

The  Area  of  Rectangles. 

Bills 50 

Miscellaneous 51  to  58 

Approximate  Answers,  Review  Problems. 

CHAPTER  II. 

Fractions 59  to  84 

Greatest  Common  Divisor,  Least  Common  Multiple, 
Addition  and  Subtraction  of  Fractions,  Cancellation, 
Ratio,  Multiplication  and  Division  of  Fractions. 

Decimals 84  to  91 

Multiplication  of  Decimals,  Division  of  Decimals. 

v 


vi  Contents. 

PAGES 

United  States  Monet ,        92  to  93 

Fractional  Parts  of  a  Dollar 

Denominate  Numbers 94  to  99 

Reduction,  Addition,  Subtraction,  Multiplication, 
and  Division. 

Measurements 99  to  101 

Areas  and  Surfaces. 

Bills 101  to  102 

Review  of  Simple  Numbers 102  to  118 

Short  Methods,  Sight  Exercises,  Sight  Approxima- 
tions, Review  Problems. 

CHAPTER  III. 

Decimals 119  to  132 

Notation  and  Numeration,  Reduction,  Addition,  Sub- 
traction, Multiplication,  Division. 
United  States  Money     .         .         .        .         .        .        .     132  to  133 

Denominate  Numbers 133  to  139 

Reduction,    Addition,    Subtraction,    Multiplication, 

Division  (two  denominations). 
Measurements 139  to  144 

Areas  of  Rectangles,  Areas  of  Right-angled  Triangles. 

Bills 144  to  145 

Percentage 145  to  147 

Interest 148  to  152 

Review  of  Simple  Numbers  and  Fractions         .         .     152  to  162 

Sight  Approximations,  Special  Drills,  Cancellation, 

Ratio,  Short  Methods,  Review  Fractions. 
Miscellaneous  Problems 162  to  172 

Oral  and  Written. 

CHAPTER  IV. 

Denominate  Numbers 173  to  189 

Reduction,  Descending  and  Ascending,  Compound 
Addition,  Subtraction,  Multiplication,  and  Division, 
Avoirdupois  Weight,  Time  between  Dates. 


Contents.         t  \s:\:  »/,  vhV  i 

Percentage       .  • 189  to  194 

Applications  and  Simple  Interest. 
Measurements  .........     195  to  209 

Area  of  Rectangles,  Square  Measure,  Solid  Contents, 

Cubic    Measure,    Surfaces    of    Rectangular    Solids, 

Angles,  Triangles,  Quadrilaterals. 
Review  op  Simple  Numbers  and  Fractions         .         .     209  to  218 
.  Special  Drills,  Sight  Approximations,  Fundamental 

Processes,   Cancellation,   Review  Fractions,  Review 

Decimals. 
Review  Problems 218  to  228 

Miscellaneous,  Oral,  Written. 

CHAPTER  V. 

Percentage 229  to  276 

Finding  Percentage,  Base,  Rate  ;  Commission,  Insur- 
ance, Duties,  Taxes,  Profit  and  Loss,  Commercial 
Discount,  Interest,  Partial  Payments,  Bank  Discount, 
Interest  by  Aliquot  Parts. 

Denominate  Numbers 277  to  291 

Reduction  Descending  and  Ascending,  Addition,  Sub- 
traction, Multiplication,  Division,  Review. 

Review  of  Simple  Numbers,  Fractions,  and  Decimals    291  to  309 


CHAPTER  VI. 

Ratio  and  Proportion ,    310  to  328 

Ratio,  Proportion,  Partitive  Proportion,  Partnership, 
Compound  Proportion. 

Involution  and  Evolution 328  to  338 

Square  Root,  Applications  of  Square  Root,  Cube  Root. 

Mensuration 339  to  357 

The  Circle,  Areas  of  Circles,  Areas  of  Triangles,  Areas 
of  Quadrilaterals,  Surfaces  of  Prisms  and  Cylinders, 
Surfaces  of  Pyramids  and  Cones,  Volumes  of  Prisms 
and  Pyramids,  Volumes  of  Cylinders  and  Cones,  Sur- 
face of  Sphere,  Volume  of  Sphere,  Circular  Measure. 


viii .  Contents. 

!    ','  <f  '<  r<     '  ',    '    ,  .  '    t  '  PAGS8 

Longitude  and  Solar  Time      .        .        .        .  .    358  to  363 

Standard  Time,  Solar  Time. 
Review  Problems       ........    363  to  366 

Miscellaneous,  Oral,  Written. 

Stocks  and  Bonds 367  to  372 

Domestic  Exchange '  .        .        .    373  to  377 

Sight  Drafts,  Time  Drafts,  Bills  of  Exchange. 
Interest .        .        .    378  to  380 

Compound  Interest,  Annual  Interest. 

Metric  System 380  to  384 

Review  Problems 384  to  414 

Special  Drills,  Review  of  Fractions,  Review  of  De- 
nominate Numbers,  Review  of  Commercial  Discount, 
Review  of  Interest,  Review  of  Bank  Discount,  Exact 
Interest,  Miscellaneous  —  Oral  and  Written. 


CHAPTER  VII. 

Algebraic  Equations  of  One  Unknown  Quantity      .     415  to  439 
Coefficients,  Clearing  of  Fractions,  Positive  and  Nega- 
tive  Quantities,   Addition,    Subtraction,    Removing 
Parentheses. 

Two  Unknown  Quantities 440  to  445 

Three  Unknown  Quantities 445  to  449 

Multiplication  and  Division 449  to  469 

Exponents,  and  Terms. 

Factoring 460  to  467 

Fractions 468  to  470 

Quadratics 471  to  479 

CHAPTER  VIII. 

Geometry 480  to  503 

Lines,  Angles,  Triangles,  Quadrilaterals,  Circles, 
Problems  in  Construction,  Calculating  Heights  and 
Distances. 


SUGGESTIONS   TO   TEACHERS. 

Additions  and  Omissions.  —  The  teacher  should  freely  supplement 
the  work  of  the  text-book  when  it  is  found  necessary  to  do  so  ;  and 
the  pupils  should  not  be  required  to  continue  the  work  under  any 
topic  after  they  fully  understand  it,  even  though  they  may  not  have 
solved  all  the  problems  given  in  connection  therewith. 

Oral  and  Written  Work. —The  heading  "Written  Problems"  is 
merely  a  general  direction,  and  it  should  be  disregarded  by  the  teacher 
when  the  pupils  are  able  to  do  the  work  "  mentally."  The  use  of  the 
pencil  should  be  required  only  so  far  as  it  may  be  necessary.  It  is  a 
pedagogical  mistake  to  insist  that  the  brighter  pupils  of  a  class  should 
set  down  a  number  of  figures  that  they  do  not  need.  As  an  occasional 
exercise,  the  pupils  may  be  directed  to  give  all  the  work  required  to 
solve  a  problem,  and  to  make  a  written  explanation  of  each  step  in 
the  solution ;  but  it  should  be  the  teacher's  aim  to  have  the  majority 
of  the  examples  done  with  as  great  rapidity  as  is  consistent  with  abso- 
lute correctness.  It  will  be  found  that,  as  a  rule,  the  quickest  workers 
are  the  most  accurate. 

Conduct  of  the  Recitation. —It  is  often  advisable,  for  some  pur- 
poses, to  divide  an  arithmetic  class  into  two  sections,  even  where  the 
pupils  are  nearly  equal  in  attainments.  The  members  of  one  sec- 
tion may  work  examples  from  their  books  while  the  others  write  the 
answers  to  oral  problems  given  by  the  teacher,  etc. 

Where  a  class  is  thus  taught  in  two  divisions,  the  members  of  each 
should  sit  in  alternate  rows,  extending  from  the  front  of  the  room  to 
the  rear.  Seated  in  this  way  each  pupil  is  doing  a  different  kind  of 
work  from  those  on  the  right  and  the  left,  and  he  does  not  have  the 
temptation  of  a  neighbor's  work  to  lead  him  to  compare  answers. 

To  save  time,  explanations  of  new  subjects  may  be  given  to  the 
whole  class  ;  but  much  of  the  arithmetic  work  should  be  done  in  "  sec- 
tions," one  of  which  is  under  the  immediate  direction  of  the  teacher, 
while  the  other  is  employed  in  "seat"  work.  The  "seat"  work  of 
pupils  of  the  more  advanced  classes  should  consist  largely  of  problems 
solved  without  assistance.    Especial  pains  have  been  taken  to  grade  the 


Suggestions  to  Teachers. 


.'PfdbjeHja  sdas-to  have' none  beyond  the  capacity  of  the  average  pupil. 
It  is  not  necessary  that  all  the  members  of  a  division  should  work  the 
same  problems  at  a  given  time,  or  the  same  number  of  problems,  or 
that  a  new  topic  should  be  postponed  until  all  of  the  previous  problems 
have  been  solved. 

Whenever  it  is  possible,  each  of  the  members  of  the  division  work- 
ing under  the  teacher's  immediate  direction  should  take  part  in  all  the 
work  done.  In  mental  arithmetic,  for  instance,  while  only  a  few  may 
be  called  upon  for  explanations,  all  of  the  pupils  should  write  the 
answers  to  each  question.  The  same  is  true  of  much  of  the  sight 
work,  the  approximations,  some  of  the  special  drills,  etc. 

Drills  and  Sight  Work.  —  To  secure  reasonable  rapidity,  it  is 
necessary  to  have  regular  and  systematic  drills.  These  should  be 
employed  frequently,  but  should  not  last  longer  than  five  or  ten 
minutes.  A  page  of  special  sight  drills  is  given  in  each  chapter. 
These  may  also  be  used  in  oral  problems. 

It  often  happens  that  as  pupils  go  forward  in  school  they  lose  much 
of  the  readiness  in  oral  and  written  work  that  they  possessed  in  the 
lower  grades,  owing  to  the  neglect  of  their  teachers  to  continue  to 
require  quick,  accurate  review  work  in  the  operations  previously 
taught.  In  this  book  these  special  drills  follow  the  plan  of  the 
combinations  of  the  earlier  book,  but  gradually  grow  more  difficult. 
They  should  first  be  used  as  sight  exercises,  either  from  the  books  or 
from  the  blackboard. 

To  secure  valuable  results  from  drill  exercises,  the  utmost  prompt- 
ness in  answers  should  be  required. 

Language.  —  While  the  use  of  correct  language  should  be  insisted 
upon  in  all  lessons,  children  should  not  be  required  in  arithmetic  to 
give  all  answers  in  "complete  sentences."  Especially  in  the  drills, 
it  is  important  that  the  results  be  expressed  in  the  fewest  possible 
words.  The  teacher  should  be  careful  always  to  employ  exact  arith- 
metical language  and  to  require  it  from  the  pupils. 

Objective  Illustrations.— The  chief  reason  for  the  use  of  objects 
in  the  study  of  arithmetic  is  to  enable  pupils  to  work  without  them. 
While  counters,  weights  and  measures,  diagrams,  or  the  like  are  neces- 
sary at  the  beginning  of  some  topics,  it  is  important  to  discontinue 
their  use  as  soon  as  the  pupil  is  able  to  proceed  without  their  aid. 

Approximate  Answers.  —  An  important  drill  is  furnished  in  the 
"approximations"  (see  Arts.  104,  180,  238,  etc.).  Pupils  should  be 
required  in  much  of  their  written  work  to  estimate  the  result  before 
beginning  to  solve  a  problem  with  the  pencil.     Besides  preventing  an 


Suggestions  to  Teachers,   ,  xT 

absurd  answer,  this  practice  will  also  have  the  effect  of  causing  'a 
pupil  to  see  what  processes  are  necessary.  In  too  many  instances, 
work  upon  a  problem  is  commenced  before  the  conditions  are  grasped ; 
this  will  be  less  likely  to  occur  in  the  case  of  one  who  has  carefully 
"estimated"  the  answer.  The  pupil  will  frequently  find,  also,  that 
he  can  obtain  the  correct  result  without  using  his  pencil. 

Indicating  Operations.  — It  is  a  good  practice  to  require  pupils  to 
indicate  by  signs  all  of  the  processes  necessary  to  the  solution  of  a 
problem,  before  performing  any  of  the  operations.  This  frequently 
enables  a  pupil  to  shorten  his  work  by  cancellation,  etc.  In  the  case 
of  problems  whose  solution  requires  tedious  processes,  some  teachers 
do  not  require  their  pupils  to  do  more  than  to  indicate  the  operations. 
It  is  to  be  feared  that  much  of  the  lack  of  facility  in  adding,  multiply- 
ing, etc.,  found  in  the  pupils  of  the  higher  classes  is  due  to  this  desire 
to  make  work  pleasant. 

Sight  Exercises.  —  Many  pupils  who  find  it  difficult  to  solve  prob- 
lems read  to  them  readily  make  the  necessary  calculations  without  a 
pencil  when  they  have  the  numbers  before  them  on  the  blackboard,  or 
in  their  books.  It  may  be  found  advisable  to  have  a  class  first  solve 
the  whole  of  a  given  set  of  oral  problems  from  their  books,  and  at  a 
later  lesson  write  the  answer  to  each  question  after  it  has  been  read 
by  the  teacher.  In  the  case  of  sight  exercises  too  difficult  to  be  solved 
mentally,  the  set  might  be  taken  up  one  at  a  time  by  individual  pupils, 
after  which  the  pupils  might  be  required  to  write  answers  "  at  sight  " 
at  a  signal  from  the  teacher.  If  the  exercises  are  on  the  blackboard, 
the  teacher  might  use  a  pointer  to  indicate  the  example  whose  answer 
was  desired,  not  following  the  order  in  which  they  appeared  on  the 
blackboard.  A  similar  method  might  be  employed  in  sight  work  done 
from  the  books. 


NEW 
GEAMMAK  SCHOOL  ARITHMETIC. 

~tto* 

CHAPTER  L 

PAGK8 

Mixed  Numbers 1  to  16 

Addition,  Subtraction,  Multiplication,  Division  (Common 

Denominators  determined  by  inspection). 
Review  of  Simple  Numbers        .        .        .        .        .        .  17  to  24 

Notation  and  Numeration,  Special  Drills,  Fundamental 

Processes. 
Decimals  (three  places) 25  to  32 

Addition  and  Subtraction,  Multiplication  and  Division 

by  an  Integer. 
United  States  Monet 32  to  43 

Fractional  Parts  of  a  Dollar,  Division  of  United  States 

Money. 
Denominate  Numbers 43  to  45 

Time,  Dry  and  Liquid  Measures,  Avoirdupois  Weight, 

Miscellaneous  Examples. 
Measurements 46  to  49 

The  Area  of  Rectangles. 

Bills 50  to  51 

Miscellaneous 51  to  58 

Approximate  Answers,  Review  Problems. 

MIXED  NUMBERS. 
1.   Preliminary  Exercises. 

How  many  halves  in  1  ?     How  many  fourths  in  1  ?     Six 
halves  =  ?     12  fourths  =  ?     6  thirds  =  ?     12  sixths  ==  ? 

f  =  ?        |  =  ?        |  =  ?        |  =  ?         ll  =  ?         Y  =  ? 

1 


d  '    '  '      "  ...  Chapter  One. 

r*«»  'A  mixed*  number  is   a  whole  number   and   a  fraction 
written  together. 

3.  A  proper  fraction  is  a  fraction  whose  numerator  is  less 
than  its  denominator. 

An  improper  fraction   is  a  fraction  whose  numerator   is 
equal  to  or  greater  than  its  denominator. 

4.  Change  each  of  the  following  improper  fractions  to  a 
whole  number  or  to  a  mixed  number : 


¥ 


¥ 


V 


¥       ¥       i 


5.   Oral  Exercises. 

How  many  quarts  in  a  gallon? 

What  part  of  a  gallon  is  a  quart  ? 

\  gallon  =  how  many  quarts  ?     \  =  how  many  fourths  ? 

How  many  quarts  in  a  peck  ?  What  part  of  a  peck  is  one 
quart?  One-half  peck  is  how  many  quarts?  One-half 
=  how  many  eighths  ? 

\  peck   is  how  many  quarts  ?   \  =  how  many  eighths  ? 


\  =how  many  eighths  ? 


how  many  eighths? 


il 


I  1 


si 


Mixed  Numbers.  3 

6.  Draw  a  line  one  foot  long.  Draw  a  second  line  of  the 
same  length ;  divide  it  into  halves.  Divide  a  third  line  of 
the  same  length  into  three  equal  parts.  Divide  three  other 
lines,  one  into  fourths,  one  into  sixths,  and  one  into  twelfths. 

How  many  inches  in  a  foot  ?  What  part  of  a  foot  is  one 
inch  ?    J  foot  =  how  many  inches  ?   \  —  how  many  twelfths  ? 

1  =  how  many  twelfths  ?  f  =  how  many  twelfths  ?  Change 
\  to  twelfths.     Change  j,  f  to  twelfths.    How  many  twelfths 

_   1  ?     2?     3  ?    4?     5?     6? 
—  $"•     3"'      6*     ^'      6*      6  ' 

2    _   ?  3    _   ?  4    _  ?  _  2 

T2"— 6  TJ  —  f  TJ  —  ^  —  ^ 

6    _   ?  _   ?  _   ?  8_?_?  9_? 

T2—  "6"—  T  —  "2  T2"—  ~5  —  T  T2~  —  4~ 

10—1  44  =  1=1  =  1=  i 

How  many  inches  in  f  ft  +  J  ft.  +  J  ft.  + 1  ft.  +  ^  ft.  ? 
How  many  feet  and  inches  ? 

How  many  12ths  in  i+i  +  i+J  +  yV?  Change  to  a 
mixed  number.  Change  the  fractional  part  to  a  different 
fraction  having  the  same  value. 

What  fraction  of  a  dime  is  1  cent  ?  £  dime  =  how  many 
cents?     £  =  TV 

£  dime  =  how  many  cents  ?  £  =  TV     Change  f  to  tenths. 

!•   t    !• 

Add  -i-  dime,  £  dime,  and  ^  dime.  How  many  cents? 
How  many  tenths  =  1  +  i  +  TV  ?  Can  y°u  change  the 
answer  to  a  different  fraction  having  the  same  value  ? 

7.  Oral  Problems. 

1.  I  spent  1  of  a  dollar  for  a  ball  and  T^  of  a  dollar  for 
a  bat.     What  part  of  a  dollar  did  I  spend  for  both  ? 

2.  What  is  the  cost  of  a  pen-knife  at  f  of  a  dollar,  and 
a  book  at  i  of  a  dollar  ? 

3.  I  need  -J  of  a  yard  of  ribbon  for  one  hat  and  £  of  a 
yard  for  another.     How  much  ribbon  should  I  buy  ? 


4  Chapter  One. 

4.  Sold  f  of  a  pound  of  tea  to  one  customer  and  £  to 
another.     How  much  was  sold  to  both  ? 

5.  What  quantity  of  oats  should  I  buy  to  give  J  of  a 
peck  to  one  horse  and  |  to  another  ? 

6.  If  I  sell  i-  of  a  dozen  of  oranges  to  one  person  and  J 
of  a  dozen  to  another  person,  what  part  of  a  dozen  do  I  sell  ? 

7.  f  of  an  hour  is  how  many  minutes  ? 

8.  I  spent  J  of  an  hour  reading  and  ^j-  of  an  hour  writ- 
ing.    What  part  of  an  hour  did  I  spend  at  both  ? 

9.  A  boy  is  carrying  6 J  pounds  of  flour,  and  6 J  pounds 
of  ham.     What  is  the  weight  of  his  load  ? 

10.   18  hours  are  what  part  of  a  day  ? 

ADDITION   OF  MIXED  NUMBERS. 

8.  In  fractions  the  numbers  above  the  line  are  called  numera- 
tors ;  the  numbers  below  the  line  are  called  denominators. 

The  numerator  and  the  denominator  are  called  the  terms  of  a 
fraction. 

To  add  fractions  they  must  have  a  common  denominator. 

A  common  denominator  is  a  number  that  will  exactly  contain 
all  the  denominators. 

The  least  common  denominator  is  the  least  number  that  will 
exactly  contain  all  the  denominators. 

9.  Add  12},  6fc  8J,  15f ,  |. 

24 


12} 

12 

61 

16 

H 

6 

15* 

20 

| 

9 

43| 

ff 

2f.     Arts.  43f. 


Mixed  Numbers.  5 

An  inspection  of  the  denominators,  2,  3,  4,  6,  8,  shows  that  24  is 
the  smallest  number  that  will  contain  each  without  remainder.  This 
is  the  least  common  denominator. 

Instead  of  writing  the  least  common  denominator  24,  with  each 
fraction,  we  may  place  it  above,  and  write  only  the  new  numerators. 
}  =  if,  |  =  ££,  i  =  fo  etc.  Write  12,  16,  6,  20,  9.  The  sum  of  these 
numerators,  63,  is  written  over  the  denominator  24,  making  the  sum  of 
the  fraction  ff.  This  improper  fraction  is  reduced  to  2|$,  and  the 
fractional  part  is  reduced  to  f.  f  is  placed  under  the  fractions  to  be 
added,  and  2  is  carried  to  the  whole  numbers,  making  43. 

Add  the  fractions  and  unite  their  sum  with  the  sum  of  the 


The  fractional  parts  of  answers  should  be  reduced  to  lowest 
terms. 


10.  Written  Exercises. 

Add: 

1.   23* 

2.    73* 

3.     93| 

4.       11$ 

5.     18* 

63* 

H 

2* 

3* 

7* 

n 

39& 

7±A 

20A 

9A 

3A 

16* 

6* 

5+ 

i 

6.   12* 

7.   19* 

8.     73} 

9.       5** 

io.  loo* 

3A 

n 

98* 

38* 

75* 

27f 

34£ 

i 

23* 

9* 

Ji- 

_± 

33* 

17* 

49* 

ll.   33| 

12.     6^ 

13.   103* 

14.   218f 

15.   444* 

m 

18* 

84* 

301* 

518f 

2*A 

32* 

25* 

18* 

37* 

69A 

94* 

H 

24| 

95| 

6  Chapter  One. 

11.   Written  Problems. 

1.  A  merchant  sold  17f  yards  of  muslin,  14J-  yards  of 
silk,  and  as  many  yards  of  calico  as  of  the  other  two 
together.     How  many  yards  did  he  sell  in  all? 

2.  A  boy  has  to  walk  from  his  home  to  a  house  If  miles 
east  of  his  home,  and  from  there  to  a  place  2£  miles  west 
of  his  home.     How  far  has  he  to  walk  ? 

3.  From  a  piece  of  cloth  17-J-  yards,  5f  yards,  and  4f 
yards  were  sold.     How  many  yards  were  sold  ? 

4.  A  man  walked  12^  miles  Tuesday,  16f  miles  Wed- 
nesday, 22  ^  miles  Thursday.  How  far  did  he  walk  in  3 
days? 

5.  A  farmer  owned  3  fields  containing,  the  first  21| 
acres,  the  second  27f  acres,  and  the  third  28^  acres.  How 
many  acres  were  there  in  all  ? 

6.  A  man  bought  3  loads  of  wood  containing  respec- 
tively 1J,  cords,  If  cords,  and  If  cords.  How  many  cords 
of  wood  did  he  buy  ? 

7.  A  man  has  10^  acres  of  wheat,  6f  acres  of  corn,  20f 
acres  of  barley,  16  §  acres  of  rye.  How  many  acres  of  grain 
has  he? 

8.  William  lives  24£  rods  from  school,  James  6^  rods 
farther  than  William,  and  Charles  10^f  rods  farther  than 
James.     How  far  does  Charles  live  from  school  ? 

9.  Henry  weighs  58T3br  pounds,  Peter  65}  pounds,  and 
John  67f  pounds,  and  their  father  as  much  as  all  three  of 
them.     How  much  does  their  father  weigh  ? 

10.  A  dealer  mixed  2-J-  pounds  of  black  tea  costing  32 
cents  per  pound  with  1£  pounds  of  green  tea  costing  40 
cents  per  pound.  How  much  per  pound  does  the  mixed  tea 
cost  him  ? 


Mixed  Numbers.  7 

SUBTRACTION  OF  MIXED  NUMBERS. 

12.  Preliminary  Exercises. 

l-i=?    li-i=?     10-*=?     10J-i=?     10i-li=? 

In  subtraction  of  mixed  numbers,  as  in  addition,  the 
fractions  must  have  a  common  denominator. 


Subtract : 

1.  left 

2.     49$$       3. 

38$f        4. 

i»A 

5.    27A 

13A 

37$$ 

29$$ 

"ft 

16ft 

6.     28^ 

7.     47$         8. 

36i$       9. 

25$$ 

10.    32$$ 

13* 

29$ 

1»A 

19$$ 

ISA 

13.   From  197$  take  68$. 

15 

Reduce  the  fractions  to  the  least 

common 

denominator 

197| 

9 

15,  as  in  addition  of  fractions.     {%  being 

greater  than 

68$ 

10 

■fs,  we  change  197-&  to  196  +  1£,  or  196f 
=  i|.     196-68  =  128.    Ans.  128}|. 

*•    tt-tt 

128$* 

14 
T5 

Reduce  the  fractions  to  the  least  common  denominator, 
and  subtract  the  fractions  and  the  integers  separately. 

14.   "Written  Exercises. 

1.     36|         2.     63L         3.     27f         4.     05f        5.    105^ 
-*i  -9TV  -17i  -25f  -8J 


6.  120$ 

7.     39$ 

8.     13$ 

9.    99$      10. 

67$ 

-84$ 

-38$ 

-7ft 

-21$ 

• 

-59$ 

11.  lOOJy 

12.  25A 

13.     93A 

14.   101$$      15. 

12$ 

76$ 

5$ 

24$ 

98$ 

4$ 

16. 


21. 


26. 


Chapter  One. 

23£   17. 

9A 

ft 

18.  133} 
27* 

19. 

16^ 

3* 

20.  37J 
29J 

52|    22. 

64£ 

18£ 

23.  125-ft 
lOOf 

24. 

47i 
8* 

25.  72TV 
50£ 

31f    27. 

27A 

63^ 
44j. 

28.   3^ 

29. 

25f 

Hi 

30.  102/j 
86J 

15.   Oral  Problems. 

1.  A  man  had  $6^,  and  he  spent  $  3  J.  How  much 
money  had  he  left  ? 

2.  Take  $  8 \  from  $  12f.  How  many  quarters  of  a 
dollar  are  there  in  the  remainder? 

3.  One-half  of  our  books  are  in  the  case ;  we  have  in  all 
184  books ;  one-half  of  the  remainder  are  on  the  table. 
How  many  are  on  the  table  ? 

4.  If  6  apples  cost  14  cents,  what  will  3  cost  ? 

5.  How  many  hours  from  10  a.m.  to  10  p.m.  ? 

6.  A  man  had  1000  acres  of  land  and  sold  996 J  acres. 
How  many  acres  had  he  left  ? 

7.  If  a  man  earns  $  14£  in  a  week,  and  spends  $  8f , 
how  much  does  he  save  ? 

8.  Bought  sugar  for  5J  cents  a  pound,  and  sold  it  for  6J 
cents  a  pound.     What  was  the  gain  on  200  pounds  ? 

9.  What  will  12f  pounds  of  beef  cost  at  16  cents  a 
pound? 

10.  If  a  girl  studies  5\  hours  in  school,  and  1\  hours  at 
home  each  day,  how  many  hours  does  she  study  in  a  week 
of  five  days  ? 


Mixed  Numbers.  9 

16.   Written  Problems. 

1.  The  weight  of  a  tub  of  butter,  including  the  weight 
of  the  tub,  is  48J  pounds.  The  tub  weighs  9£  lb.  What  is 
the  butter  worth  at  24  cents  per  pound  ? 

2.  A  farmer  had  7  bushels  of  potatoes.  He  used  2 
bushels  and  3  pecks  for  seed.  What  would  the  remainder 
be  worth  at  20  cents  per  peck  ? 

3.  How  much  heavier  is  a  cheese  weighing  40J  pounds 
than  one  which  weighs  26f  pounds  ? 

4.  A  farmer  having  217  bushels  of  corn  sold  95 }{ 
bushels j  how  many  bushels  had  he  left? 

5.  A  milliner  gained  1-J  dollars  by  selling  a  hat  for  6£ 
dollars ;  what  did  it  cost  her  ? 

6.  From  a  cask  of  oil  containing  43f  gallons,  17|  gallons 
were  drawn ;  how  many  gallons  remained  ? 

7.  A  man  having  25J  dollars  paid  6J  dollars  for  coal, 
2\  dollars  for  dry  goods,  and  £  of  a  dollar  for  a  pound  of 
tea ;  how  much  had  he  left  ? 

8.  A  butcher  buys  an  ox  weighing  alive  1200  pounds, 
at  6  cents  per  pound.  When  killed  and  dressed,  its  weight 
is  f  of  the  live  weight.  What  is  the  butcher's  profit,  if  he 
sells  the  meat  at  an  average  of  15  cents  per  pound  ? 

9.  A  farmer  sold  36^  dozen  eggs  to  one  storekeeper,  5J 
dozen  to  another,  17f  dozen  to  a  third,  8f  dozen  to  a  fourth, 
and  11T72-  dozen  to  a  fifth.  How  much  did  he  receive  for 
them  at  12  cents  per  dozen  ? 

10.  A  teacher's  salary  per  month  is  135^  dollars,  and 
his  expenses  average  51 J  dollars :  how  much  does  he  save 
per  month  ? 

11.  A  man  gave  \  of  his  money  to  his  wife  and  i  of  it  to 
his  daughter.  He  divided  the  remainder  equally  among  his 
three  sons,  each  of  whom  received  $1000.  How  much 
money  had  he? 


io  Chapter  One. 

MULTIPLICATION  OF  MIXED  NUMBERS. 

17.  Preliminary  Exercises. 

i  +  i  +  }  =  ?  3timesJ  =  ?  ixS  =  ? 

6  times  \  =  ?  3  times  £  =  ?  |x3  =  ? 

|X9  =  ?  |xl5  =  ?  |xl7  =  ? 

|X7  =  ?  f  x  20  =  ?  £  x  12  =  ? 

£X5  =  ?  fxl3  =  ?  |xlO  =  ? 

18.  Multiplication  of  a  mixed  number  by  an  integer. 

Find  the  product  of  235f  by  39. 

235f 

Multiply  3  by  39;   divide  the  result  by  4:  the  39 

quotient,  29£,  is  39  times  f .    Write  the  next  partial      4)117 
product,  235  x  9  ;  then  the  product  of  135  by  3  tens.  29J 

The  sum  of  the  three  partial  products  gives  the        2115 


result,  9 194 J. 

705 

19.   Oral  Exercises. 

9194J  Ans 

1.  1|X9  =  ?          3. 

3|x5  =  ? 

5.   5fxl2  =  ? 

2.    2|x7=?           4. 

4|  x  8  =  ? 

6.    6^x10  =  ? 

20.   Written  Exercises. 

1.   215§xl7  =  ?      3. 

417^x20  =  ? 

5.  619^x19  =  ? 

2.   316fxl5  =  ?      4. 

518£xl3=? 

6.  720T\x23  =  ? 

7.        163|                 9. 
X75 

509£ 
X213 

11.       6089f 
X1004 

8.        103|               10. 
Xl7 

308| 
X156 

12.       1607f 
X2340 

Mixed  Numbers.  n 


21.  Multiplication  of  an  integer  by  a  mixed  number. 
Multiply  276  by  280|. 


276 

280# 
Multiply  276  by  the  numerator,  3;    divide   the     Q\eo8 
product  by  the  denominator,  8  ;  the  quotient,  103£,  is       ^.n4  ^ 
the  product  of  276  by  f .     Multiply  276  by  8  tens  and 
by  2  hundreds,  etc. 


2208 
552 
77383^  Ans. 

To  multiply  a  whole  number  by  a  fraction,  place  the  product 
of  the  numerator  by  the  whole  number  over  the  denominator, 
and  reduce,  if  possible. 

22.  Written  Exercises. 

1.  13x7|   =?       4.  17x10^  =  ?  7.  102x22f  =  ? 

2.  19x8^  =  ?   5.  21xll|  =?  8.  204x34f  =  ? 
6.  27  x  12f  m  ?  9.  468  x  56f  =  ? 

12.   4060      14.   3579TV 
X  2050|         x  4300 


3. 

23  x  9^-  = 

9 

10. 

387 
x400f 

11. 

698 
Xl35f 

13.   3050      15.   4987^- 
X  2060f         x  2469 


23.  Oral  Problems. 

1.  How  many  ounces  in  6%  pounds  ? 

2.  I  sold  3^  yards  of  silk  and  2f  yards  of  velvet.     How 
many  yards  in  all  did  I  sell  ? 

3.  From  60  take  24.     Find  \  of  the  remainder. 

4.  |  of  100  rods  =  ?  6.   |  of  81  yards  =  ? 

5.  (fof60)-9  =  ?  7.   f  of  56  pounds  =  ? 


12  Chapter  One. 

8.  |  of  a  yard  and  12  inches  are  how  many  inches  ? 

9.  If  one-half  a  pound  of  soap  costs  10  cents,  what  will 
three  pounds  cost  ? 

10.  John  is  going  a  journey  of  100  miles ;  if  he  travels 
|  of  the  distance  in  the  cars  and  the  rest  in  a  coach,  how 
many  miles  will  he  travel  in  the  coach  ? 

11.  How  many  times  must  I  fill  my  glass,  which  holds 
£  a  pint,  to  fill  my  pitcher,  which  holds  a  gallon  ? 

12.  If  a  boy  is  in  school  h\  hours  a  day,  how  many 
hours  is  he  in  school  in  200  days  ? 

24.   Written  Problems. 

1.  What  is  meant  by  f  of  any  number  or  thing  ?  Make 
a  drawing  to  show  what  you  mean. 

2.  What  is  the  cost  of  15^-  acres  of  land  at  $45  an  acre  ? 

3.  Reduce  £,  f,  §,  and  ^  to  fractions  having  a  common 
denominator. 

4.  What  is  the  cost  of  a  side  of  beef  containing  252 
pounds  at  9J  cents  a  pound  ? 

5.  A  hotel  uses  18f  pounds  of  beef  in  a  day.  What  will 
be  the  weekly  bill  at  22  cents  a  pound  ? 

6.  A  man  walks  3^  miles  in  one  hour.  How  far  can  he 
walk  in  9  hours  ? 

7.  From  a  piece  of  muslin  containing  37£  yards,  three 
pieces  each  measuring  7-J-  yards  were  sold.  How  much 
remained  in  the  piece  ? 

8.  At  $  7.86  a  barrel,  what  will  18|  barrels  of  flour  cost  ? 

9.  Bought  6  bushels  of  apples  at  62 \  cents  a  bushel, 
and  sold  them  at  12£  cents  a  half-peck.  What  was  the 
gain? 

10.  In  a  school  containing  945  pupils  ^  of  the  number 
were  boys ;  how  many  boys  in  the  school  ? 

11.  What  is  the  cost  of  15  acres  of  land  at  $  45£  an  acre  ? 


Mixed  Numbers.  13 

12.  If  a  quart  of  cream  is  worth  22  cents,  what  are  two 
gallons  worth  ? 

13.  At  9  cents  a  quart,  what  is  the  cost  of  2\  gallons 
of  vinegar  ? 

14.  What  is  the  total  quantity  of  molasses  in  4  casks 
containing,  respectively,  40 J,  25f  27-^-,  and  55|  gallons  ? 

15.  The  Post-office  Department  bought  6670  pounds  of 
twine  at  19 J  cents  a  pound ;  372  pounds  of  sponge  at  65£ 
cents  a  pound,  and  40£  dozen  of  ink  at  $2  a  dozen. 
What  was  the  total  cost  of  the  purchase  ? 


DIVISION  OF  MIXED  NUMBERS. 

25.   Preliminary  Exercises.  t , 

How  many  times  is  \  of  a  dollar  contained  in  $  1  ?  How 
many  times  is  1  of  a  pint  contained  in  1  pint  ?  \  of  a  gal- 
lon in  1  gallon  ? 

How  many  times  is  ^  of  a  dollar  contained  in  $  2  ?  In  $  3  ? 
In  $5? 

How  many  times  is  -J-  of  a  dollar  contained  in  $1.50? 
In  $2.50?     In  $3.50?     In  $4.50? 

How  many  times  is  1  half  contained  in  3  halves  ?  In 
5  halves  ?    In  7  halves  ?     In  9  halves  ? 

3-1-1=?  5^_\—9  l-±-l  =  ?  5.-^1  =  ? 

2-Y*  2     '     2  '  2     '    2  *  T    *    ^         ' 

How  many  times  is  f  contained  in  f  ?    In  f  ?    In  -^  ? 

In^f? 
Divide  1|  by  If    4£  by  If     7|  by  If     10J  by  If 
Divide  3  by  If     6  by  If    9  by  If    12  by  If    15  by  If 
Divide  5  by  If    6Jbylf    10  by  If    lljbylf    15  by  If 
Divide  i  by  f.      £by|.      1}  by  f .      1$  by  f.      2J  by  $ 

3  by  J.    3fby{. 


14  Chapter  One. 

26.  Written  Exercises. 

1.  Divide  250  by  12$. 

250  =  500  halves.        12$  =  25  halves. 
600  halves  -=-  25  halves  =  500  ^  25  =  20,  Ans. 
Proof  :  20  x  12$  =  250. 

2.  Divide  62\  by  25. 

62$  =  125  halves.        25  =  50  halves. 

125  halves  -r-  50  halves  =  125  -r-  50  =  2£f  =  2$,  Ans. 

Proof  :  25  x  2$  =  62$. 

3.  Divide  1387$  by  18}.  Ans. 74 

18|  =  75  fourths.  75)5550 

Change  1387 1  to  fourths  by  multiplying  by  4.  ^25 

1387$  x  4  =  5550 ;  that  is,  1387$  =  5550  fourths.  300 

75  fourths  is  contained  in  5550  fourths  74  times.  300 

Reduce  the  dividend  and  the  divisor  to  improper  fractions  of 
the  same  denominator,  and  divide  the  numerator  of  the  divi- 
dend by  the  numerator  of  the  divisor.  Prove  the  correctness 
of  the  answer  by  multiplying  the  quotient  by  the  divisor. 

27.  Written  Exercises. 


Div 
1. 

ide: 
60+  \ 

11. 

75  + 12$- 

21. 

62|  +  12$. 

2. 

60  +  1* 

12. 

150  + 12$- 

22. 

187$ +  12^ 

3. 

60+  i 

13. 

75+  6J 

23. 

81 J  +  6J 

4. 

60  +  1* 

14. 

150+  6J 

24. 

193}+  6J 

5. 

60+  \ 

15. 

62  +  15$- 

25. 

77$-  + 15$ 

6. 

60  + If 

16. 

105  +  17$. 

26. 

192^  +  17$ 

7. 

60+  * 

17. 

69+  5} 

27. 

97}+  5} 

8. 

60 +  2* 

18. 

93+  7} 

28. 

193}+  7} 

9. 

60+  } 

19. 

100  +  33$ 

29. 

166$  +  33$ 

10. 

60  +  3J 

20. 

150  +  16f 

30. 

133$  + 16  $ 

Mixed  Numbers.  15 

31.  601-^2       34.  87i-f-6J      37.  60  -- 3f 

32.  60 

33.  15f 


71      35.  621 --6J       38.  241 
If      36.  60f-=-3       39.  87£ 


If 


28.   Oral  Problems. 

1.  I  paid  18  cents  for  11  pounds  of  lard.     What  is  the 

price  per  pound  ? 

36  cents  for  3  pounds. 

2.  At  f  dollar  per  yard,  how  many  yards  of  silk  can  be 
bought  for  $  9  ? 

36  quarter  dollars  -=-  3  quarter  dollars. 

3.  If  one  fish  cost  25  cents,  how  much  would  2\  fish 
cost? 

4.  A  man  bought  30  apples  at  the  rate  of  3  for  5  cents. 
How  much  did  he  give  for  them  ? 

5.  If  I  pay  6  cents  for  a  dozen  apples,  how  much  does 
each  apple  cost  ? 

6.  How  many  times  is  41  contained  in  27  ? 

7.  If  21  bushels  of  oats  will  keep  a  horse  one  week,  how 
long  will  18  bushels  keep  him  ? 

8.  If  $  97  is  \  of  a  sum  of  money,  what  is  that  sum  ? 

9.  What  is  the  cost  of  12  doz.  eggs  at  the  rate  of  2  eggs 
for  3  cents  ? 

10.  If  3  boys  can  cut  a  cord  of  wood  in  8  hours,  how 
long  will  it  take  4  boys  to  cut  a  cord  ? 

11.  If  i  of  a  melon  costs  15  cents,  what  will  two  melons 
cost  at  the  same  rate  ? 

12.  It  takes  2\  yards  of   cloth  for  a  pair  of   trousers. 
How  many  pairs  can  be  made  from  30  yards  of  cloth  ? 

13.  Paid  $  12.90  for  3  pieces  of  lace.     How  much  did 
each  cost  ? 

14.  If  3  straw  hats  cost  63  cents,  what  will  be  the  cost 
of  5? 


1 6  Chapter  One. 

29.   Written  Problems, 

1.  A  farmer  distributed  15  bushels  of  corn  among  several 
persons,  giving  them  If  bushels  apiece;  among  how  many- 
persons  did  he  divide  it  ? 

2.  A  man  bequeathed  to  his  son  $  3500,  which  was  -f-  of 
what  he  left  his  wife.     How  much  did  he  leave  his  wife  ? 

Suggestion.  —  }  of  wife's  share  =  $  3500.     Multiplying  by  7  : 
5  times  wife's  share  =  $24,500. 

3.  If  f  of  a  farm  is  valued  at  $  1728,  what  is  the  value 
of  the  whole  ? 

4.  A  man  walks  4f  miles  in  one  hour,  how  far  can  he 
walk  in  9  hours  ? 

5.  At  |  of  a  cent  a  foot,  how  many  feet  of  wire  can  be 
bought  for  $1.26? 

6.  The  sum  of  69f  dollars  was  divided  equally  among 
5  men ;  what  was  each  one's  share  ? 

7.  At  |  dollars  per  yard,  how  many  yards  of  cloth  can 
be  purchased  for  $98? 

8.  In  how  many  days  can  a  horse  eat  66  bushels  of  oats 
if  he  eats  f  of  a  bushel  a  day  ? 

9.  A  man  bought  chairs  at  4f  dollars  apiece  for  114 
dollars,  and  then  sold  them  at  6  J  dollars  apiece ;  how  much 
did  he  gain? 

10.  A  man  sold  9f  bushels  of  seed  for  $61.60;  find  the 
price  per  bushel. 

11.  What  part  of  24  is  3  ?     What  part  of  24J  is  8}  ? 

12.  What  would  be  the  cost  of  24^  pounds  of  beans  at 
the  rate  of  11  cents  for  3£  pounds  ? 


Review  of  Notation  and  Numeration.        17 

30.  Notation  and  Numeration. 

The  largest  number  that  can  be  written  with  six  figures 
is  999,999. 

1,000,000,  is  called  one  million. 

Write  in  figures  two  million.  Three  million.  Four 
million.     Six  million.     Eight  million.     Ten  million. 

31.  Bead  the  following : 

1.  1,234,567  6.  11,034,065  11.  30,100,021 

2.  3,000,560  7.  14,602,500  12.  35,000,600 

3.  5,009,008  8.  17,386,925  13.  401,023,160 

4.  7,090,070  9.  20,007,316  14.  760,030,020 

5.  9,843,000  10.  25,000,005  15.  980,750,000 

32.  Write  in  figures : 

1.  Seventy-eight  million,  one  hundred  eight  thousand, 
ninety-six. 

2.  Three  million,  eight. 

3.  Fourteen  million,  seven  thousand,  five. 

4.  Nine   hundred   eighty-seven   thousand,  six  hundred 
fifty-four. 

5.  Twenty  million,  thirty  thousand,  forty. 

6.  Three  hundred  seven  million,  nine  hundred  four  thou- 
sand, six. 

7.  Nine   hundred    ninety-nine   million,   nine    hundred 
ninety-nine  thousand,  nine  hundred  ninety-nine. 

8.  Four  hundred   seventy-six    million,   three   hundred 
thousand. 

9.  Thirty-four  thousand,  eighteen. 

10.  Sixty-four  million,  thirty-two  thousand,  sixteen. 

11.  Add  the  foregoing. 


1 8  Chapter  One. 


REVIEW  OF  FUNDAMENTAL  OPERATIONS. 

Practice  in  the  fundamental  operations  should  not  be  neglected. 
Business  men  complain  that  elementary  and  high  school  graduates 
cannot  add. 

Read  the  following  numbers.     Add  each  column. 


1. 

27,083,549              2.    508,900,007 

3.    2 

43,576,908 

3,006,005                       4,629,880 

5,987,600 

20,080,070                     25,936,097 

380,070 

1,647,893                   134,870,603 

68,000 

206,045                     59,009,300 

593,056 

73,000                       7,000,004 

2,384,672 

180,059                          686,909 

59,876,004 

4. 

9,256,874             5.                348 

6. 

7,293 

863,052                             2,967 

82,538 

24,635,998                          36,847 

786,324 

7,007,007                         243,837 

94,649 

6,875,634                        183,634 

256,834 

3,987,456                         986,246 

3,983,387 

35,068                            8,216 

54,619 

705                        586,237 

760,888 

33.   Oral  Exercises. 

Give  answers : 

1. 

1200x6       6.   1300x9      11.   2100x4 

16. 

1400x8 

2. 

1800x4       7.   2300x3      12.    1400x6 

17. 

2400  x  4 

3. 

2500x3       8.   3200x2      13.   4100x2 

18. 

1300  x  7 

4. 

1700x5       9.    1500x4      14.   1600x5 

19. 

1200  x  9 

5.   1400x7     10.    1200x8      15.    2200x3      20.   6300x2 


Review  of  Fundamental  Operations.         19 


34.   Written  Exercises. 
Multiply: 

1.  9,207x3014 

2.  5,482  x  798f 

3.  5,290  x  6075 

4.  9,204  x  678J 

5.  75,074  x  395 

6.  68,431  x  924| 


7.  95  x  95  x  95 

8.  185  x  19  x  78 

9.  87£x  23  x  36 

10.  706  x  304x509 

11.  48Jx  32  x  74 

12.  538  x  247  x  125 


35.   Oral  Exercise 

«. 

Divide : 

1.      960-240 

6. 

8400-^2100 

11. 

10800-1200 

2.      780  h- 260 

7. 

8600-4300 

12. 

10400 -j- 1300 

3.      960 -5- 480 

8. 

8800-2200 

13. 

6000 -r- 1500 

4.      720-180 

9. 

9600-3200 

14. 

5700-1900 

5.    2170-310 

10. 

9900 -;- 3300 

15. 

12000 -h  2400 

The  foregoing  exercises  are  given  as  a  preparation  for  the  long 
division  drill  that  follows.  Each  of  the  above  set  has  an  exact  quo- 
tient, easily  determined  at  sight. 

The  object  of  the  following  set  is  to  drill  pupils  to  obtain  rapidly 
the  correct  quotient  figure  in  a  long  division  example.  A  pupil  giving 
4  as  the  answer  to  No.  1  should  be  asked  to  give  the  product  of  241  by  4. 

36.   Long  division  drill.     (Omit  remainders.) 


1.  960-241 

2.  779^-260 

3.  959-480 

4.  720-181 

5.  1160 -r- 130 


6.  8,400-2110 

7.  8,500-4300 

8.  8,800-2199 

9.  9,599-3199 
10.  10,000 -j- 3330 


11.  10,800-1205 

12.  10,300-1300 

13.  6,100-^1550 

14.  5,700  - 1899 

15.  12,020-5-2410 


<zo  Chapter  One. 

37.  Divide: 

1.  34,463-5-370  7.   703,705 -j- 12,345 

2.  823,150-1298  8.  420,135-  6,789 

3.  639,712-624  9.  370,088 -j-  5,986 

4.  345,738^-7210  10.  510,940-=-  4,900 

5.  861,704-351  11.  639,215-  9,783 

6.  857,384-3004  12.  345,678 -j-  7,095 

38.  Sight  Exercises. 

Note. — First,  combine  the  quantities  within  the  parentheses,  (  ); 
next,  complete  the  combinations  within  the  brackets,  [  ],  if  any ;  then 
perform  the  remaining  operations. 

28  +  (40  -j-  2)  =  28  +  20 
[30  --  (6  -4-  2)]  x  5  =  [30  --  3]  x  5  =  10  x  5 

Perform  indicated  operations  at  sight : 

1.  18  +  (30x4)  7.  }  of  (240  +  60) 

2.  7  +  (2x8)-4  8.  (7  +  2)x(8-4) 

3.  [(7  +  2)x8]-4  9.  7  +  [2x(8-4)] 

4.  l-(i  +  i)  10.  1-i  +  i 

5.  (6xJ)  +  i  ii.  6x(i+i) 

6.  \  of  \  of  600  12.  jxl2x| 

SPECIAL  DRILLS. 

Note.  —  In  adding,  subtracting,  and  multiplying  without  using  the 
pencil,  it  is  inadvisable  to  begin  with  the  units :  68  and  34,  for  in- 
stance, are  more  readily  combined  mentally,  by  adding  68  and  30  (88) 
and  4.     In  the  recitation,  the  pupil  should  say  88,  92 ;  or  92,  merely. 

630  +  280  =  630  +  200  +  80. 

39.  Give  sums : 

56  +  25  32  +  48  750  +  190  225  +  54 

47+47  29  +  28  390  +  120  315  +  21 

22  +  68  65  +  26  480  +  150  437  +  60 


Review  of  Fundamental  Operations.  21 

40.  Give  remainders : 

92  -  58  =  92  -  50  -  8.     Say  42,  34. 
840  -  280  =  840  -  200  -  80.     Say  640,  560. 

81-56              750-190  750-560  279-54 

94-47              510-120  510-390  386-63 

60-28              630-150  630-480  457-37 

72-39              820-160  820-660  568-25 

41.  Give  products : 

87  x  2  =  (80  x  2)  +  (7  x  2).     Say,  160,  174. 

410x6                83x7  43  x    5  12x70 

310x9               99x2  26  x    7  18x30 

420  x  4                65  x  3  24  x    8  16  x  40 

630x3               49x4  22  x    9  13x50 

740x2                37x5  18x11  11x60 

42.  Give  quotients : 

168-3      168-56  1470-7  1470-210 

196-4      196-49  2790-^-9  2790-^310 

190-5      190-^38  1680-5-4  1680  -*-  420 

192-6      192-32  1890 -r- 3  1890-630 

196 -r- 7      196-*- 28  1480 -h  2  1480 -s- 740 

43.  Give  answers : 

2^  +  lJ           If--   {  fof  66  12J+| 

2J  +  1J           2}-H  84  x     f  Si^-f 

2£  +  l£           8|-2|  foflOO  51  +  f 

21  +  li           H-H  186x     i  3|  +  f 

2J  +  H           6J-4J  |ofl20  4£  +  | 


11  Chapter  One. 

44,   Oral  Problems. 

1.  Paid  59^  for  muslin  and  25^  for  trimming.  How 
much,  was  paid  for  both  ? 

2.  A  boy  had  75^.  How  much  had  he  after  spending 
25^  for  a  knife  and  15^  for  a  ball  ? 

3.  If  8  pounds  of  raisins  cost  $  1.04,  what  is  the  price 
per  pound  ? 

4.  At  $  1.89  per  yard  of  silk,  what  will  be  the  cost  of 
J  of  a  yard  ? 

5.  If  32  pounds  of  flour  cost  96  cents,  how  many  pounds 
can  be  bought  for  60  cents  ? 

6.  One  girl  has  16  cents,  another  has  24  cents,  a  third 
has  8  cents.  How  many  dolls  at  16  cents  each  can  be 
bought  with  their  money  ? 

7.  What  will  be  the  weight  of  3  bushels  of  corn,  weighing 
56  pounds  per  bushel  ? 

8.  How  many  ounces  in  9  pounds  avoirdupois  ? 

9.  How  many  pounds  in  8  packages,  each  weighing  10 
ounces  ? 

10.  Find  the  cost  of  3  pounds  and  2  ounces  of  butter  at 
32  cents  per  pound. 

11.  Bought  4  pounds  of  sugar  at  6  cents  a  pound,  and  a 
pound  of  butter  at  36  cents.     How  much  change  from  $  1  ? 

12.  Four  boys  have  144  marbles  among  them.      If  the 
marbles  were  equally  divided,  how  many  would  each  have  ? 

13.  A  man   earns   $100   per   month,  and  spends  $76. 
How  much  does  he  save  ? 

14.  If  a  man  saves  $  32  per  month,  how  many  months 
will  it  take  him  to  save  $  960  ? 

15.  Paid  $27.90  for  9  jackets.      What  did  they  cost 
apiece  ? 


Review  of  Fundamental  Operations.         23 

16.  Mr.  B's  farm  contains  520  acres.     How  many  acres 
will  he  have  left  after  selling  180  acres  ? 

17.  William's  kite  string  is  435  yards  long,  John's  is  62 
yards  longer.     What  is  the  length  of  John's  string  ? 

18.  A  farmer  raised  168  bushels  of  grain.     He  sold  £  of 
it.     How  many  bushels  did  he  sell  ? 

19.  A  piece  of  ribbon  measuring  6 \  yards  is  cut  into  pieces 
a  quarter  of  a  yard  long.     How  many  pieces  are  there  ? 

20.  If  it  takes  18f  yards  of  cloth  to  make  3  suits,  how 
many  yards  does  it  take  for  1  suit  ? 

21.  James  has   150   marbles,  Thomas   has  -f  as  many. 
How  many  marbles  have  both  ? 

22.  A  newsdealer  received  $6.36  for  papers  sold  at  3 
cents  each.     How  many  papers  did  he  sell  ? 

23.  If  it  takes  4|-  days  for  one  man  to  do  a  piece  of  work, 
how  long  will  it  take  2  men  to  do  the  same  work  ? 

24.  A  farm  is  divided  into  4  fields,  each  containing  49 
acres.     How  many  acres  are  there  in  the  farm  ? 

25.  From  a  piece  of  cloth  containing  10 \  yards,  5f  yards 
are  sold.     How  many  yards  are  left? 

26.  Find  the  cost  of  28  pounds  coffee  at  $  \  per  pound. 

27.  How  much  does  a  farmer  receive  for  28  cows  which 
he  sells  at  $30  each? 

28.  Find  the  number  of  hours  in  a  week. 

29.  How  many  pieces,  each  three-quarters  of  a  yard  long, 
can  be  cut  from  six  yards  of  wire  ? 

30.  3600  seconds  are  equal  to  how  many  minutes  ? 

31.  If  25  yards  of  material  are  needed  for  a  dress,  how 
many  yards  will  be  required  for  30  dresses  ? 

32.  At  7  for  a  cent,  what  will  98  marbles  cost  ? 


24  Chapter  One. 

45.  "Written  Problems. 

1.  The  sum  of  three  numbers  is  150.     Two  of  the  num- 
bers are  68  and  43.     What  is  the  third  ? 

68  +  43  -f  ?  =  150 

2.  The  divisor  is  24;   the  dividend  is  264.     Find  the 
quotient. 

3.  The  product  is  228;  the  multiplicand  is  19.     What 
is  the  multiplier  ?  19  x  ?  =  228 

4.  The  minuend  is  175;  the  subtrahend  is  87.     What  is 
the  remainder? 

5.  The  remainder  is  92;   the  subtrahend  is  89.     Find 
the  minuend.  ?  —  89  =  92 

6.  The  minuend  is  176,  and  the  remainder  is  99.    What 
is  the  subtrahend  ? 

7.  The  multiplier  is  15;  the  multiplicand  is  46.     What 
is  the  product? 

8.  The  multiplier  is  16;  the  product  is  272.     What  is 
the  multiplicand? 

9.  The  dividend  is  300;   the  divisor  is  17.     Find  the 
remainder. 

10.  The  quotient  is  15 ;  the  remainder  is  3 ;  the  divisor 
is  8.    What  is  the  dividend? 

8)  P 
15| 

11.  The  dividend  is  273 ;  the  quotient  is  21.     What  is 
the  divisor  ? 

12.  The  dividend  is  267;  the  quotient  is  13;  the  remain- 
der is  7.     What  is  the  divisor? 

?)267 
13| 

13.  How  many  acres  of  land  could  you  buy  for  $  76,225, 
if  one  acre  cost  $37? 


Decimals.  25 


NOTATION  OF  DECIMALS. 

46.  A  decimal  fraction  is  one  in  which  the  unit  is  divided 
into  tenths,  hundredths,  thousandths,  etc. 

47.  Preliminary  Exercises. 

In  the  number  25;  what  does  the  2  stand  for  ? 

In  the  number  467,  what  does  the  4  represent  ?  The  6  ? 
The  7? 

In  the  number  33,333,  give  the  value  of  the  first  3  (com- 
mencing at  the  left).  Of  the  second.  Of  the  third.  Of 
the  fourth.     Of  the  fifth. 

The  last  3  is  what  part  of  the  number  represented  by  the 
fourth  3  ?  The  third  3  is  what  part  of  the  second  ?  Each 
3  is  what  part  of  the  3  to  its  left  ?  Upon  what  does  the 
value  of  each  3  in  this  number  depend  ? 

In  the  number  XXXIII,  what  is  the  value  of  the  first  X  ? 
Of  the  second  ?     Of  the  third  ? 

When  we  write  $784,365,  the  7  stands  for  seven  times 
how  many  dollars  ?  The  8  for  eight  times  how  many  dol- 
lars? The  4  for  four  times  how  many  dollars?  The  3 
stands  for  three  times  what  part  of  a  dollar  ?  The  6  stands 
for  six  times  what  part  of  a  dollar  ?  The  5  stands  for  five 
times  what  part  of  a  dollar  ? 

Hundreds.    Tens.    Units.    Decimal  Point.  Tenths.    Hundredths.    Thousandths. 

7  8        4  .36 

784.365  is  read  784  and  365  thousandths. 
37.5  is  read  37  and  5  tenths. 
6.492  is  read  6  and  492  thousandths. 
400.75  is  read  400  units  and  75  hundredths. 

Note.  —  In  reading  a  number  containing  an  integer  and  a  decimal, 
the  word  and  may  be  placed  between  the  two,  as  is  shown  above. 
To  avoid  mistakes,  the  word  units  should  be  used  after  the  integer 
in  reading  such  numbers  as  200.005.  Say :  Two  hundred  units  and 
five  thousandths. 


16  Chapter  One. 

48.  Read  the  following : 

1.  .7  5.     3.275  9.   100.025 

2.  34.9  6.   32.4  10.         .125 

3.  .36  7.     1.025  11.         .005 

4.  .95  8.       .35  12.       1.348 

49.  Express  in  decimals : 

1.  7  tenths. 

2.  36  and  47  thousandths. 

3.  One  hundred  twenty -five  thousandths. 

4.  One  hundred  units  and  twenty-five  thousandths. 

5.  47  hundredths. 

6.  Four  hundred  units  and  six  tenths. 

7.  Four  hundred  six  thousandths. 

8.  3  and  56  hundredths. 

9.  65  hundredths. 
10.  6  and  5  tenths. 

Note.  —  Since  ffo  equals  ^,  .50  =  .5.  The  cipher  at  the  right  of 
.60  has,  therefore,  no  value,  •fflfo  =  ^  ;  .700  is,  therefore,  the  same 
as  .7.    In  giving  answers,  reject  ciphers  at  the  right  of  the  decimal. 

ADDITION  OF  DECIMALS. 


50.  Add: 

1.  .7 

2.  3.84 

3.  28.978 

4.  5.6 

4.18 

68.075 

.28 

.387 

.005 

.5 

5.375 

26.93 

5.67 

24.698 

18.758 

8.754 

10.555  97.113 

Write  the  numbers  so  that  the  decimal  points  stand  in  a 
column.  Add  as  in  whole  numbers,  and  place  the  point  in 
the  sum  directly  under  the  points  in  the  addends. 


Decimals.  Y] 

51.  Written  Exercises. 

1.  .027  +  1.39  +  48.6  +  72.978   ' 

2.  234.96  +  .675  +  50.4  +  6.02  + 1.001 

3.  3.047  +  54.79  +  .097  +  .76  +  .862 

4.  .8  +  .38  +  .479  +  27.87  +  375 

5.  .445 +  34.75 +  306.973 +  .004 +  48.56 

6.  .81  + 12.654  +  234.79  +  8.6  +  .603  +  42.96 

7.  45.78  +  .237  +  6.987  + 18  +  372.003  +  37.5 

8.  4.745  +  36.58  +  725.894  +  9.87  +  75.357  +  86.74 

9.  59.3  +  83  +  9.64  +  48.565  +  6.98  +  8.795  +  963.826 

10.  13.387  +  72.563  +  .7  +  .603  +  7.245  +  .483  +  9.25 

11.  8.3  +  2.576  +  3.42  + 1.5  +  6.279  +  .003  + 1.417 

SUBTRACTION  OF  DECIMALS. 

52.  From  37  take  3.7. 

37  may  be  written  37.0 
subtract    3.7 

33.3  Ans. 
In  practice,  the  pupil  should  not  waste  time  in  writing  the  unneces- 
sary ciphers  at  the  right  of  the  decimals  in  the  minuend. 

182.01  1.  28.6 

-4.624  -.009  -1.003 

177.386  .991  27.597 

Write  the  numbers  so  that  the  decimal  point  in  the  subtra- 
hend stands  directly  under  the  decimal  point  in  the  minuend. 
Subtract  as  in  whole  numbers,  and  place  the  point  in  the  remain- 
der under  the  points  above. 


53.   Written  Exercises. 

Find  answers : 

1.   1-.057                3.    6-3.324 

5.   3-1.568 

2.   1-.245                4.   4-2.491 

6.   7-4.736 

28  Chapter  One. 

7.  3.587-1.34  14.   681.38-94.572 

8.  91.352-72.456  15.   1000-465.874 

9.  42.007-17.658  16.   30.053-18.7 

10.  68.217-39.4  17.  2568.91-1925 

11.  9.34-5.672  18.  1.234  -  .825 

12.  45.268-23.068  19.  473.5-298.572 

13.  219.843-187.95  20.  57.083-44.95 

MULTIPLICATION  OF  A  DECIMAL  BY  AN  INTEGER. 
54.  Three  times  3  tenths  equals  how  many  tenths  ? 
.3x3  =  what?  .3x4  =  ?  .3x12  =  ? 

1.  Multiply  2.7  by  8. 

8  times  7  tenths  =  56  tenths  =  5.6.     Write  .6.     8  times         \ 

2  =  16;carry5-  Mb.   21S 

2.  Multiply  .275  by  12.  275 
The  product  of  275  thousandths  by  12  is  3300  thousandths,      X  12 

which  equals  3  and  300  thousandths,  or  3  and  3  tenths.  3.300 

Ans.  3.3 

Multiply  as  in  whole  numbers,  and  point  off  in  the  product 
decimal  places  equal  to  the  number  in  the  multiplicand,  reject- 
ing unnecessary  ciphers  at  the  right  of  the  decimal. 


55.  "Written  Exercises. 

Multiply : 

1.     .36  x  3 

6. 

.048  x  375 

2.   57.2x7 

7. 

12.67  x  300 

3.     6.4  x  122 

8. 

6.57  x  9 

4.      .67  x  4 

9. 

8.76  x  43 

5.   38.4x25 

10. 

005  x  360 

Decimals.  29 

56.  Oral  Exercises. 
Give  products : 

1.  6.84x10  6.  .961x100 

2.  68.4x10  7.     .57x1000 

3.  3.28x10  8.     .09x1000 

4.  5.71x100  9.  .026x100 

5.  5.71x1000  *  10.  5.17x10 

Note.  — The  pupil  should  deduce  the  rule  for  multiplying  a  decimal 
by  10,  100,  1000. 

57.  To  multiply  an  integer  by  a  decimal. 
Multiply  35  by  6.4. 

35  6.4 

"•4  Since  the  product  of  35  by  6.4  is  equal  to  the  ^5 

14.0        product  of  6.4  by  35,  there  will  be  one  decimal  32.0 

210  place  in  the  product.  192 

224.0  Ana.    224.  224.0 

In  multiplying  an  integer  by  a  decimal,  or  a  decimal  by 
an  integer,  point  off  in  the  product  as  many  decimal  places 
as  there  are  decimal  places  in  the  multiplier  or  the  mul- 
tiplicand. 

58.  Multiply: 

1.  122  by  6.4  6.   5430  by  .8 

2.  512  by  .003  7.     748  by  .97 

3.  .056  by  987  8.     964  by  .347 

4.  97  by  .005  9.     570  by  .11 

5.  275  by  1.2  10.     570  by  1.1 


30  Chapter  One. 

DIVISION  OF  A  DECIMAL  BY  AN  INTEGER. 
59.   Preliminary  Exercises. 


1.     8.64 

4-2 

6.    .6664-6 

2.   48.24 

-=-4 

7.   .048-8 

3.     .465 

-4 

8.     .814-9 

4.     8.40 

4-5 

9.     .12  4-5 

.   5.        8.4 

4-5 

10.      .34  4-4 

60.   Where  it  is  necessary,  ciphers  may  be  annexed  to  the  right  of 
the  decimal  in  the  dividend. 

1.    8)  .12 
.015 

2. 

15).06 
.004 

1.875 
5.   64)120. 
64 

.012 

.413 

56.0 

3.   125)1.50 
1.25 

4. 

21)8.673 
8.4 

51.2 
4.80 

.250 

.27 

4.48 

.250 

.21 

.320 

.063 

.320 

.063 
In  dividing  a  decimal  by  an  integer,  point  off  in  the  quotient 
as  many  decimal  places  as  there  are  decimal  places  in  the 
dividend  (including  the  ciphers  annexed). 

Note.  —  In  practice,  however,  the  decimal  point  may  be  placed  in 
the  quotient  under  (or  over)  the  decimal  point  in  the  dividend. 

61.  "Written  Exercises. 
Divide : 

1.  25)1.00  6.  11)70.07 

2.  4)21.80  7.  24)36J5 

3.  8]^  8.  18p76 

4.  13)3.913  9.  25)TET 

5.  12)48.12  10.  32)62.000 


Decimals. 

62.   Perform  the  indicated  divisions : 

A  =  '  +  25           25)1.00 

1.           i  = 

6. 

TTS"  — 

2.           i  = 

7. 

w- 

3.           |  = 

8. 

w- 

4-        A  = 

9. 

H^= 

5.          |  = 

10. 

A  = 

63.   Give  quotients  at  sight : 

1.   932-100 

6. 

684-100 

2.     86-1000 

7. 

57.6-*- 10 

3.   328-10 

8. 

24.3-100 

4.       9^1000 

9. 

8.75-*- 10 

5.     48-1000 

10. 

932.5-^100 

31 


Note.  —  The  pupil  should  deduce  the  rule  for  dividing  by  10,  100, 
1000. 

64.   Written  Problems. 

1.  A  man  had  10.5  yards  of  cloth,  and  used  4.125  yards 
to  make  a  coat.     How  many  yards  did  he  have  left  ? 

2.  Find  the  cost  of  2.578  acres  of  land,  at  $  37  an  acre. 

3.  Find  the  amount  of  .87  and  8.7.  Find  the  difference 
between  .906  and  90.6. 

4.  Write  in  figures :  Seventy-six  thousand  four  hundred 
nine,  and  eighty-two  thousandths.  Nine  hundred  thousand 
nine  hundred  units,  and  thirty-one  hundredths. 

5.  A  franc  is  19.3  cents.  Find  the  cost  in  United  States 
money  of  goods  amounting  to  1250  francs. 

6.  A  merchant  bought  1800  meters  of  silk.  How  many 
yards  did  he  buy,  a  meter  being  39.37  inches  ? 


3  2  Chapter  One. 

7.  A  kilogram  is  2.2046  pounds.  What  is  the  difference 
in  weight  between  the  English  ton  of  2240  pounds  and  a 
French  ton  of  1000  kilograms  ? 

8.  A  cubic  foot  of  water  weighs  1000  ounces.  How  many- 
pounds  does  a  cubic  foot  of  gold  weigh,  gold  being  19.4 
times  as  heavy  as  water  ? 

9.  There  are  128  cubic  feet  in  a  cord.  How  many  tons 
of  2000  pounds  are  there  in  a  cord  of  pine  wood,  the  latter 
being  .66  times  as  heavy  as  water? 

10.  A  man  buys  three  plots  of  ground  containing  35.27, 
17.8,  and  40.375  acres,  respectively.  Find  the  total  cost  at 
$36  per  acre. 

11.  How  many  pints  are  there  in  2.375  gallons  ? 

12.  What  decimal  of  a  peck  is  a  quart  ? 

13.  What  will  be  the  cost  of  carrying  468  tons  of  coal  at 
$0,125  per  ton? 

14.  A  farmer  sold  one-eighth  of  his  farm  of  224.2  acres 
at  $  62.50  per  acre.     How  much  did  he  receive  for  it  ? 

UNITED  STATES   MONEY. 
65.  Learn  the  following  table : 

10  mills  =    1  cent. 
100  cents  =    1  dollar. 


1  dime  =  10  cents. 
1  eagle  =  10  dollars. 

ADDITION  AND  SUBTRACTION  OF  UNITED  STATES  MONEY. 

66.   Add  the  following  without  placing  the  amounts  in 
columns : 

1.  $8.34,  $40.39,  $638.27,  $594.38,  $1.97. 

2.  $0.03,  $8.05,  $600.00,  $38.72,  $198.52,  $0.63. 


United  States  Money.  23 

3.  $432.84,  $96.25,  $3.64,  $782.46,  $800.06,  $6.50. 

4.  $3.60,  $40.05,  $91.86,  $350.48,  $84.00,  $287.63. 

5.  $98.27,  $0.60,  $600.39,  $8.09,  $37.38,  $503.07. 

6.  $202.97,  $42.23,  $453.60,  $7.18,  $63.54,  $0.37. 

7.  $8.43,  $0.54,  $2.57,  $85.13,  $425.31,  $8.27. 

8.  $486.54,  $84.62,  $1.96,  $8.13,  $35.84,  $236.49. 

9.  $83.61,  $523.00,  $23.04,  $0.86,  $35.82,  $584.60. 
10.  $34.80,  $93.54,  $200.41,  $324.86,  $50.14,  $8.75. 

The  foregoing  examples  may  be  added  directly  from  this  book 
or  from  the  blackboard,  the  pupils  writing  on  their  slates  or  papers 
nothing  but  the  answers. 

67.  Subtract  the  following  without  rearranging  them. 
Find  the  sum  of  the  minuends,  the  sum  of  the  subtrahends, 
and  the  sum  of  the  remainders. 

1.  $1,000.00-     $876.49  = 

2.  $549.37-      $99.89  = 

3.  $345.93-       $76.04  = 

4.  $1786.08-  $1097.19  = 

5.  $345.00-     $187.23  = 


6.  ?          —  T          = 

7.  $3545.37-     $966.38  = 

8.  $82.46-        $7.59  = 

9.  $5074.02-  $4987.63  = 

10.  $77.84-        $9.88  = 

11.  $4680.35-  $4679.46  = 


12.  ?         - 


34  Chapter  One. 

MULTIPLICATION  OF  UNITED  STATES  MONEY. 

68.  Find  the  cost  of: 

1.  197  barrels  of  flour,  at  $  5.66  per  barrel. 

2.  486  bushels  of  wheat,  at  $  1.04  per  bushel 

3.  237  tons  of  plaster,  at  $  6.72  per  ton. 

4.  809  tons  of  hay,  at  $  11.45  per  ton. 

5.  74  carloads  of  bran,  at  $  20.62^  per  load. 

6.  208  sheep,  at  $  4.65  per  head. 

7.  673  barrels  of  mackerel,  at  $  10.60  per  barrel. 

8.  984  bushels  of  onions,  at  $  1.09  per  bushel. 

9.  99  pounds  of  butter,  at  24  cents  per  pound. 

10.  208  pounds  of  coffee,  at  28  cents  per  pound. 

69.  What  will  be  the  cost  of  157  pounds  of  sugar,  at  5p 
per  pound. 

At  5^  per  pound  157  pounds  will  cost  157  times  5^.    In  157 

practice,  however,  we  multiply  157  by  the  smaller  number  5.  5 

Ans.  $7.85.  jgg 

11.  1376  yards  of  muslin,  at  6f  £ 

12.  2084  bushels  of  corn,  at  47^. 

13.  1864  pounds  of  beef,  at  5|£ 

14.  988  pounds  of  turkeys,  at  13|£ 

15.  296  bushels  of  potatoes,  at  47|£ 

16.  1272  pounds  of  dried  apples,  at  2f# 

17.  488  pounds  of  lard,  at  10J f. 

18.  2240  pounds  of  sugar,  at  4|^. 

19.  5176  pounds  of  wool,  at  30|y. 

20.  4892  bushels  of  wheat,  at  99f  £ 


United  States  Money.  35 

DIVISION  OF  UNITED  STATES  MONEY. 

70.  Oral  Exercises. 

How  often  is  1  quart  contained  in  1  gallon  ?  1  pint  in 
1  quart  ?  2  quarts  in  1  gallon  ?  1  inch  in  1  foot  ?  2  inches 
in  1  foot  ?  3  inches  in  1  foot  ?  4  inches  in  1  foot  ?  6  inches 
in  1  foot  ?  6  inches  in  2  feet  ?  8  inches  in  2  feet  ?  1  ounce 
in  1  pound  ?  1  ounce  in  2  pounds  ?  4  ounces  in  2  pounds  ? 
1  fourth  in  1  half  ?     1  third  in  1  ? 

How  often  is  1  cent  contained  in  $  1  ?  2  cents  in  a  dollar  ? 
4  cents  in  2  dollars  ?    25  cents  in  25  dollars  ? 

$25  =  2500j* ;  2500/*  -4-  25^  =  100,  Ans. 

Note.  — When  the  divisor  is  a  concrete  number,  i.e.  a  number  asso- 
ciated with  objects,  the  dividend  must  be  a  like  concrete  number ;  in 
which  case  the  quotient  will  be  an  abstract  number,  i.e.  a  mere  number. 

3  dollars,  4  coats,  7  apples,  are  concrete  numbers ;  3,  4,  7,  are  ab- 
stract numbers. 

When  the  divisor  is  abstract  and  the  dividend  concrete,  the  quotient 
is  concrete. 

71.  Give  answers  at  sight: 

1.  $4-10^  11.  $1-*-^ 

8.  95-f-  5l  12.  $3-=-$  J 

3.  $12-*-  4^  13.  $84-5-50^ 

4.  $86-+-   6^  14.  $l-=-16^ 

5.  $63 -j-   3?  15.  $16 -=-16^ 

6.  $7-f-25^  16.   $16-*-16# 

7.  $20-33^  17.  $16 -=-33^ 

8.  $36-=-   3^  18.  $16-=-25^ 

9.  $40-=-50j*  19.  $16-50^ 
10.     $9-f-10^  20.  $12-f-20^ 


36  Chapter  One. 

72.  At  36  cents  each,  how  many  spellers  can  be  bought 

for  $27? 

75 

$  27  =  2700  cents.    Since  1  speller  costs  36  cents,  the     Qayrmf) 

number  of  spellers  that  can  be  bought  for  2700  cents  will  be  oro 

2700  -  36  =  75.  Ana.  75  spellers.  ^-r 

180 

-IQf) 

73.  Written  Problems.  — 

1.  At  $  2.75  per  day,  how  long  will  it  take  a  man  to  earn 
$110?  (11,000-*- 275.) 

2.  How  many  yards  of  muslin,  at  12  cents  per  yard,  can 
be  bought  for  $  126? 

.3.   A  farmer  spent  $  140  for  sheep  at  $  5.60  each.     How 
many  did  he  buy  ? 

4.  A  grocer  pays  $  74.50  for  tea  at  %  of  a  dollar  per 
pound.     What  is  the  weight  of  the  tea  ? 

5.  When  rye  is  worth  87  cents  per  bushel,  how  many 
bushels  can  be  purchased  for  $  261  ? 

6.  At  12}  cents  per  pound,  how  many  pounds  of  meat 
will  cost  $  175.25  ? 

7.  If  75  spellers  cost  $  27,  what  is  the  price  of  1  speller? 

If  75  spellers  cost  $  27,  1  speller  will  cost  ^  of  $  27. 

75)$  27.00 

The  divisor  75  is  an  abstract  number.     The  dividend  being  a  con- 
crete number,  the  quotient  will  be  concrete,  viz.  $  .36. 

8.  A  woman  paid  $  24  for  36  yards  of  dress  goods.    What 
did  she  pay  per  yard  ? 

9.  At  6  for  a  dollar,  how  many  rabbits  can  be  bought  for 

$87? 

10.    The  cost  of  13  houses  was  $36,887.50.    What  was 
the  price  of  each  ? 


United  States  Money.  37 

FRACTIONAL  PARTS  OF  A  DOLLAR. 

SHORT  METHODS. 

74.  What  will  be  the  cost  of  16  base-balls  at  25  cents 
each? 

At  $1  each,  16  base-balls  cost  16  quarter-dollars,  or  $4. 

75.  Oral  Exercises. 

At  25  cents  per  pound,  yard,  dozen,  etc.,  what  will  be  paid 
for: 

1.  32  base-balls  ?  7.  37  dozen  lemons  ? 

2.  52  pounds  coffee  ?  8.  25  bushels  tomatoes  ? 

3.  48  straw  hats  ?  9.  41  panes  of  glass  ? 

4.  84  yards  ribbon  ?  10.  33  packages  of  candy  ? 

5.  36  second  readers  ?  11.  49  Koman  candles  ? 

6.  56  vases  ?  12.  60  bars  of  soap  ? 

76.  At  50  cents,  give  the  cost  of: 

1.  46  pounds  tea.  7.   76  grammars. 

2.  28  pairs  of  scissors.  8.   57  boxes  of  pens. 
3..  38  penknives.  9.   49  picture  books. 

4.  84  third  readers.  10.   83  dolls. 

5.  44  pounds  candy.  11.   27  games. 

6.  32  caps.  12.   75  feather  dusters. 

77.  How  many  cents  in  one-eighth  of  a  dollar  ? 

At  one-eighth  of  a  dollar  each,  what  will  be  the  cost  of 
24  bars  soap  ? 

At  $1  each,  24  bars  cost  $  2/,  or  $3. 


38  Chapter  One. 

Give  the  cost  of  the  following  at  12£  cents  per  pound,  etc. 

(ft): 

1.  16  pounds  meat.  5.  80  jars  of  jelly. 

2.  48  dozen  eggs.  6.  96  cans  of  condensed  milk. 

3.  72  straw  hats.  7.  104  yards  sheeting. 

4.  64  gallons  oil.  8.  88  pounds  currants. 

78.  How  many  cents  in  one-third  of  a  dollar  ? 

At  one-third  of  a  dollar  each,  what  will  be  the  cost  of  12 
bottles  of  cologne  ? 

At  $  l  each,  12  bottles  cost  $  >/,  0r  $4. 

Give  the  cost,  at  33^  cents  per  yard,  pound,  etc.,  of: 
1.   36  yards  of  ribbon.  4.   27  bushels  of  oats. 

£.    63  pairs  of  cuffs.  5.   54  pecks  of  walnuts. 

3.   48  pounds  of  butter.  6.    72  dozen  oranges. 

79.  How  many  cents  in  three-fourths  of  a  dollar  ? 
If  sleds  cost  $  |  each,  what  is  paid  for  16  sleds  ? 

At  $£  each,  16  sleds  would  cost  $^,  or  $4 ;  at  $f  each,  the  cost 
is  3  times  $4,  or  $12. 

Give  the  cost  of  the  following  at  75  cents  per  yard,  etc. : 

1.  48  yards  silk.  4.   28  gallons  syrup. 

2.  24  bushels  peaches.  5.   36  base-balls. 

3.  84  pounds  tea.  6.   32  concert  tickets. 

80.  Find  the  cost  of  13  pairs  of  gloves  at  75  cents  per 
pair. 

Since  13  is  not  exactly  divisible  by  4,  this  problem  should  be  handled 
as  follows : 

13  pairs  at  $£  per  pair  cost  $-^,  or  $0},  or  $9.75. 

Give  the  cost  of  the  following  at  75  cents  per  bushel,  etc.  i 

1.  11  bushels  rye.  4.     7  mats. 

2.  15  gallons  ice-cream.  5.   21  bushels  potatoes. 

3.  9  cloth  caps.  6.   18  pairs  of  skates. 


United  States  Money. 


39 


81.  Parts  of  a  Dollar. 

6£  cents  =  &  of  $  1 

8£  cents  =  ^  of  $  1 
12^  cents  =  }  of  $1 
16|  cents  =  i  of  $1 
25    cents  =  \  of  $1 
33$  cents  =  i  of  $1 

82.  Oral  Exercises. 

Give  the  cost  of  72  articles  at 

1.  12  J  cents  each. 

2.  33^  cents  each. 

3.  16|  cents  each. 


37£  cents  =  |  of  $  1 
50  cents  -\  of  $1 
62£  cents  =  £  of  $1 
66f  cents  =  $  of  $1 
75  cents  =  £  of  $  1 
87£  cents  =  |  of  $1 


4.  25    cents  each. 

5.  50    cents  each. 

6.  37 J  cents  each. 


37£  cents  =  $  f .    At  $  \  each,  the  cost  of  72  articles  would  be  $  9 ; 
at$|,  $27. 


7.  62-J-  cents  each. 

8.  87 J  cents  each. 

83.  Multiply: 

1.  6J  cents  x    16 

2.  8|  cents  x    24 

3.  12i  cents  x    88 

4.  16|  cents  x    54 

5.  25    cents  x  240 

6.  33J  cents  x    66 

7.  50    cents  x  186 

8.  37J  cents  x    48 

9.  62^  cents  x    32 


9.   66|  cents  each. 
10.   75    cents  each. 


10.  66|  cents  x    33 

11.  75    cents  x  128 

12.  87£  cents  x    88 

13.  $1.33£x  24 

14.  9*112}  X   16 

15.  $2.25    x   12 

16.  $3.75    x   12 

17.  $4.37£x     8 

18.  95.16}  x     6 


40  Chapter  One. 

84.   Find  the  cost  of: 

1.  86  neckties,  at  50  cents  each. 

2.  Six  dozen  handkerchiefs,  at  25  cents  apiece. 

3.  32  yards  of  silk,  at  $1.12J  per  yard  ($4). 

4.  64  arithmetics,  at  75  cents  each. 

5.  84  geographies,  at  $1.25  each  (f  1J). 

6.  96  pounds  of  tea,  at  75  cents  a  pound. 

7.  84  pairs  of  gloves,  at  $  1.50  per  pair. 

8.  72  yards  of  cloth,  at  $  2.12^  per  yard. 


85.   Written  Exercises. 

Note.  —  Pupils  should  be  taught  to  perform  operations  without 
placing  the  numbers  under  each  other.  In  working  examples  1  to  8, 
one  figure  is  written  at  a  time,  beginning  at  the  right.  The  answers 
to  examples  9  to  12  are  found  by  division,  one  figure  being  written  at 
a  time.  In  examples  13  to  20,  the  cents  should  be  changed  to  fractions 
of  a  dollar. 


Write  answers. 

1.  687  pounds,  at  4^. 

2.  976  yards,  at  6^. 

3.  938  coats,  at  $  7. 

4.  695  pounds,  at  20£ 

5.  12  bushels,  at  $  1.43. 

6.  11  sheep,  at  $  7.47. 

7.  9  tons,  at  $22.75. 

8.  13  sacks  of  salt,  at  $  1.11. 

9.  352  yards,  at  12^. 
10.  1728  hats,  at  50£ 


11.  933  yards,  at  33^. 

12.  2504  dolls,  at  25^ 

13.  248  pounds,  at  75^. 

14.  186  pounds,  at  66%#. 

15.  8  barrels,  at  $  16.37£. 

16.  16  gallons,  at  $3.62£. 

17.  124  bushels,  at  $  1.50. 

18.  96  pounds,  at  $1.25. 

19.  120  gallons  at  $  2.33f 

20.  64  sacks,  at  $1.12 J. 


United  States  Money.  41 

86.  Oral  Exercises. 

At  50  cents  each,  how  many  penknives  can  be  bought  for 
fl?     For  $2?     For  $3?     For  $10?     For  $  20  ? 

At  25  cents  each,  how  many  readers  can  be  bought  for 
$1?     For  $2?     For  $3?     For  $10?     For  $20? 

At  12£  cents  per  yard,  how  many  yards  can  be  bought  for 
$1?     For  $2?     For  $3?     For  $10?     For  $20? 

At  33J  cents  per  pound,  how  many  pounds  can  be  bought 
£or$l?     For  $2?     For  $3?     For  $10?     For  $20? 

87.  At  25  cents  each  (four  for  $  1)  : 

1.  How  many  base-balls  can  be  bought  for  $9  ? 

2.  Straw  hats,  for  $12? 

3.  Koman  candles,  for  $18? 

4.  Readers,  for  $  15  ? 

5.  Vases,  for  $21? 

6.  Bars  of  soap,  for  $3J? 

7.  Packages  of  candy,  for  $4£? 

8.  Yards  of  ribbon,  for  $5.75  ? 

9.  Bushels  of  tomatoes,  for  $10.50? 

10.  Pounds  of  coffee,  for  $12.75  ? 

88.  At  50  cents  (two  for  $1)  : 

11.  Pounds  of  tea,  for  $  43  ? 

12.  Penknives,  for  $20.50  ? 

13.  Pounds  of  candy,  for  $94  ? 

14.  Third  readers,  for  $  17.50  ? 

15.  Caps,  for  $  21  ? 

16.  Grammars,  for  $37? 


42  Chapter  One. 

17.  Boxes  of  pens,  for  $72? 

18.  Dolls,  for  $64? 

19.  Pairs  of  scissors,  for  $  19  ? 

20.  Feather  dusters,  for  $26.50? 

89.  At  121  cents  (eight  for  $  1)  : 

21.  Gallons  of  oil,  for  $8? 

22.  Dozen  of  eggs,  for  $  11  ? 

23.  Pounds  of  meat,  for  $21  ? 

24.  Quarts  of  plums,  for  $  1  J-  ? 

25.  Jars  of  jelly,  for  $f  ? 

26.  Yards  of  sheeting,  for  $1J? 

27.  Cans  of  milk,  for  $2J? 

28.  Pounds  of  currants,  for  $3.12£? 

29.  Whisk  brooms,  for  $4,371? 

30.  Collars,  for  $5.62J? 

90.  At  331  cents  (three  for  $1)  : 

31.  Yards  of  ribbon,  for  $  6  ? 

32.  Pairs  of  cuffs,  for  $  12  ? 

33.  Pounds  of  butter,  for  $  18  ? 
84.   Bushels  of  oats,  for  $32? 

35.  Pecks  of  walnuts,  for  $1J  ? 

36.  Dozen  of  oranges,  for  $  1$  ? 

37.  Straw  hats,  for  $2.33£? 

38.  Dolls,  for  $3.66f? 

39.  Penknives,  for  $4.33 J? 

40.  Pounds  of  candy,  for  $5.66}? 


Denominate  Numbers. 


43 


91.  At  16f  cents  (six  for  $  1)  : 

41.  Collars,  for  $4? 

42.  Pounds,  for  $  21  ? 

43.  Yards,  for  $  J? 

44.  Ounces,  for  f±  ? 

45.  Packages,  for  66 1  cents  ? 


92.  Oral  Exercises. 
Divide  at  sight : 

51.  $  24.50  by  50  cents. 

52.  9  12.25  by  25  cents. 

53.  $  26  by  33J  cents. 

54.  9  14.50  by  121  cents. 

55.  $  17  by  16|  cents. 


46.  Quarts,  for  f  1.16}? 

47.  Gallons,  for  $1.50? 

48.  Pecks,  for  f  2§T 

49.  Feet,  for  $  3.33^  ? 

50.  Yards,  for  $  4.66|  ? 


56.  $  18.75  by  25  cents. 

57.  $  11.87$-  by  12£  cents. 

58.  $  13.33J-  by  33J  cents. 

59.  $  37.50  by  50  cents. 

60.  $  13.33 1  by  16f  cents. 


DENOMINATE  NUMBERS. 
93.   Learn  the  following  tables : 

TIME. 

60  seconds  (sec.)  =  1  minute  (min.) 

60  minutes  =  1  hour  (hr.) 

24  hours  =  1  day  (da.) 

7  days  =  1  week  (wk.) 

AVOIRDUPOIS  WEIGHT. 

16  ounces  (oz.)  =  1  pound  (lb.) 

2000  pounds  =  1  ton  (T.) 

The  hundredweight  (100  pounds)  is  written  cwt. 


DRY  MEASURE. 


2  pints  (pt.) 
8  quarts 
4  pecks 


=  1  quart  (qt.) 
=  1  peck  (pk.) 
=  1  bushel  (bu.) 


44  Chapter  One. 

LIQUID  MEASURE. 

2  pints  (pt.)  =  1  quart  (qt.) 

4  quarts  =  1  gallon  (gal.) 

A  gill  (gi.)  is  equal  to  one- fourth  of  a  pint.     It  is  very  rarely  used. 

LINEAR  MEASURE. 

12  inches  (in.)  =  1  foot  (ft.) 

3  feet  =  1  yard  (yd.) 
5 £  yards                      =  1  rod  (rd.) 

320  rods  =  1  mile  (mi.) 

1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft.  =  63,360  in. 

A  furlong  is  equal  to  40  rods,  }  mile. 

A  hand,  used  in  measuring  the  height  of  horses,  =  4  in.  A  knot, 
used  in  measuring  distances  at  sea,  =  1.15  mi.  A  fathom,  used  in 
measuring  the  depth  of  the  sea,  =  6  ft. 

SQUARE    MEASURE. 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 

9  square  feet  =  1  square  yard  (sq.  yd.) 

30 \  square  yards  =  1  square  rod  (sq.  rd.) 

160  square  rods  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 

1  A.  =  160  sq.  rd.  =  4840  sq.  yd.  =  43,560  sq.  ft. 

A  section  of  land  is  a  mile  square. 

Roofing,  flooring,  and  slating  are  often  estimated  by  the  square, 
which  contains  100  square  feet. 

94.   Written  Exercises. 

1.  How  many  hours  in  7  J-  days  ? 

2.  How  many  hours  in  7  days  12  hours  ? 

3.  How  many  minutes  in  2  hours?    How  many  seconds ? 

4.  A  man  buys  12  bushels  and  3  pecks  apples  at  $  1  per 
bushel.     What  is  the  cost  ? 


Denominate  Numbers.  45 

5.  What  will  be  the  cost  of  3  pecks  7  quarts  chestnuts 
at  8  cents  per  quart  ? 

6.  How  many  pints  are  there  in  5  gallons  of  ice-cream  ? 

7.  How  many  half-pints  are  there  in  10  gallons  of  ice- 
cream ? 

8.  How  many  4-ounce  packages  can  be  made  from  5 
pounds  of  pepper? 

9.  A  boy  pays  $  1.50  for  1  gallon  and  2  quarts  of  ice- 
cream.    What  is  the  price  per  quart  ? 

10.  How  many  gallons  of  lemonade  will  be  needed  to 
give  96  people  £  pint  each? 

11.  How  many  seconds  in  5  hours? 

12.  How  many  hours  in  1  we^k  ? 

13.  Change  13  hours  and  20  minutes  to  minutes. 

14.  Change  15  bushels  4  pecks  to  pecks. 

15.  How  many  ounces  in  47  pounds  5  ounces  ? 

16.  How  many  pounds  and  ounces  in  237  ounces  ? 

17.  Change  1494  minutes  to  hours  and  minutes. 

18.  Find  the  number  of  hours  in  6  weeks. 

19.  Change  60  pounds  to  the  decimal  of  a  hundredweight. 

20.  How  many  inches  are  there  in  12  feet  2  inches  ? 

21.  How  many  pounds  in  14i  tons  ? 

22.  How  many  pounds  in  f  of  a  ton  ? 

23.  What  will  400  pounds  of  coal  cost  at  $5  per  ton? 

24.  What  decimal  of  a  ton  is  1500  pounds  ? 

25.  How  many  days  and  hours  in  £  of  a  week  ? 

26.  Find  the  number  of  yards  in  3  pieces  of  cloth,  each 
containing  16  yards  2  feet. 

27.  When  coal  is  $7.50  per  ton,  what  will  be  the  cost  of 
3000  pounds  ? 


46  Chapter  One. 

MEASUREMENTS. 
95.   Preliminary  Exercises. 


One  square  inch. 


Draw  a  square  each  side  of  which 
is  one  inch.  This  is  called  a  square 
inch.  Cut  out  of  paper  several  one-inch 
squares. 

Draw  a  rectangle  two  inches  long,  one 
inch  wide.  How  many  paper  one-inch 
squares  will  exactly  cover  it  ? 

Draw  a  rectangle  three  inches  long,  two  inches  wide. 
Divide  it  into  one-inch  squares.  How  many  one-inch 
squares  are  there  in  the  lower  row?  How  many  rows? 
How  many  square  inches  in  the  rectangle  ? 

How  many  square  inches  in  a  rectangle  6  inches  long,  3 
inches  wide  ? 

How  many  square  inches  in  a  rectangle  4  inches  long, 
4  inches  wide  ? 

How  many  square  inches  are  there  in  a  rectangle  12 
inches  long,  3  inches  wide  ?  In  a  rectangle  1  foot  long, 
3  inches  wide  ?  In  a  rectangle  1  foot  1  inch  long,  4  inches 
wide? 

Note.  —  The  foregoing  exercises  should  be  accompanied  by  accu- 
rate drawings  on  paper  or  on  the  blackboard,  which  should  lead  the 
pupils  to  see  that  the  unit  in  the  given  examples  is  the  square  inch. 
They  should  be  made  aware  that  the  number  of  squares  in  the  lower 
row  corresponds  to  the  length  of  the  rectangle  in  inches;  and  that 
the  number  of  rows  corresponds  to  the  width  of  the  rectangle.  From 
this  they  should  deduce  the  rule  : 

The  number  of  square  inches  in  the  surface  of  a  rectangle  is 
equal  to  the  number  of  inches  in  its  length  taken  as  many  times 
as  there  are  inches  in  its  width. 

This  product  is  called  the  area  of  the  rectangle. 


Measurements.  47 

96.  The  area  of  a  surface  is  the  number  of  times  that  it 
contains  another  surface,  taken  as  the  unit  of  measurement. 
Thus,  the  statement  that  the  area  of  a  surface  is  8  square 
inches  means  that  a  square  inch  is  contained  in  the  surface 
8  times. 

97.  The  sum  of  all  the  sides  of  a  figure  is  called  its 
perimeter. 

98.  Written  Exercises. 

Find  the  area  of  each  of  the  following  rectangles  in  square 
inches.     Find  the  perimeter  of  each  in  feet  and  inches. 

1.  13  in.  by  14  in.  7.  13  in.  by  42  in. 

2.  17  in.  by    9  in.  8.  27  in.  by  31  in. 

3.  18  in.  by    7  in.  9.  18  in.  by  22  in. 

4.  23  in.  by  15  in.  10.  64  in.  by  29  in. 

5.  21  in.  by  19  in.  11.  1  ft.  by  7  in. 

6.  37  in.  by  14  in.  12.  1  ft.  1  in.  by  11  in. 
Note.  —  Change  each  dimension  to  inches  before  multiplying. 

13.  1  ft.  3  in.  by  12  in.  17.  2  ft.  6  in.  by  1  ft.  3  in. 

14.  1  ft.  by  1  ft.  18.  3  ft.  7  in.  by  2  ft.  9  in. 

15.  1  ft.  4  in.  by  1  ft.  19.  4  ft.  11  in.  by  1  ft.  8  in. 

16.  2  ft.  6  in.  by  1  ft.  20.  5  ft.  3  in.  by  2  ft.  11  in. 

99.  -Oral  Exercises. 

How  many  square  feet  in  a  rectangle  2  feet  long,  1  foot 
wide? 

How  many  square  feet  in  a  rectangle  6  feet  long  by  5  feet 
wide? 

How  many  square  feet  in  a  rectangle  9  feet  long  by  7  feet 
wide  ? 

Note.  —  The  unit  in  the  following  examples  is  the  square  foot. 


48  Chapter  One. 

100.  "Written  Exercises. 

Find  the  area  in  square  feet  of  each  of  the  following  rec- 
tangles.    Find  the  perimeter  of  each  in  feet. 

1.  12  ft.  by  14  ft.  6.   29  ft.  by  12  ft. 

2.  15  ft.  by  17  ft.  7.   15J.  ft.  by  12  ft. 

3.  19  ft.  by  11  ft.  8.    15  ft.  6  in.  by  12  ft. 

4.  23  ft.  by  15  ft.  9.   18|  ft.  by  16  ft. 

5.  18  ft.  by  16  ft.  10.    18  ft.  9  in.  by  16  ft. 
Note.  — Change  the  inches  to  fractions  of  a  foot. 

11.  23J  ft.  by  18  ft.  16.  36  ft.  by  23  ft.  5  in. 

12.  24  ft.  8  in.  by  18  ft.  17.  13  ft.  by  24J  ft. 

13.  19  ft.  3  in.  by  16  ft.  18.  13  ft.  4  in.  by  24  ft. 

14.  24  ft.  by  17  ft.  9  in.  19.  26  ft.  8  in.  by  15  ft. 

15.  24  ft.  by  16  ft.  1  in.  20.  \2\  ft.  by  12  ft. 

101.  Suggestive  Examples. 

1.  Measure  the  top  of  the  desk,  disregarding  fractions 
of  an  inch,  and  calculate  the  surface  in  square  inches. 

2.  Measure  the  blackboard,  and  find  how  many  square 
feet  in  its  surface.     (Do  not  include  fractions  of  a  foot.) 

3.  Calculate  the  number  of  square  inches  in  a  pane  of 
glass  in  the  schoolroom  window. 

4.  Find  the  number  of  square  feet  in  the  floor  of  the 
classroom. 

5.  Find  the  number  of  square  feet  in  the  classroom 
ceiling. 

6.  Estimate  the  height  of  the  classroom,  and  calculate 
the  number  of  square  feet  in  the  front  wall.  7.  In  the  rear 
wall.     8.  In  the  right-hand  wall.     9.  In  the  left-hand  wall. 


Measurements.  49 

102.  Written  Problems. 

Suggestion.  —  When  the  surface  is  required  in  square  inches, 
change  each  dimension  to  inches ;  when  required  in  square  feet,  ex- 
press each  dimension  in  feet,  or  in  feet  and  the  fraction  of  a  foot ; 
when  required  in  square  yards,  etc. ,  express  each  dimension  in  yards, 
etc. 

1.  How  many  square  feet  are  there  in  the  surface  of  a 
field  125  feet  long,  87.5  feet  wide  ? 

(1  square  foot  x  125  x  87.5.) 

2.  A  rug  is  2  yards  long,  If  yards  wide.  How  many 
square  yards  does  it  contain  ? 

(1  square  yard  x  2  x  If.) 

3.  How  many  square  yards  are  there  in  a  strip  of  carpet 
6  yards  long,  27  inches  (f  yard)  wide  ? 

4.  Find  the  number  of  square  meters  in  a  room  12  meters 
long,  9.75  meters  wide. 

5.  At  50  cents  per  square  yard,  what  will  be  the  cost  of 
the  oil-cloth  needed  to  cover  a  floor  18  feet  (6  yards)  long, 
15  feet  (5  yards)  wide  ? 

6.  What  will  be  the  cost,  at  $  1.50  per  square  yard,  of 
carpeting  a  room  6^-  yards  long,  15  feet  wide  ? 

7.  At  3  cents  a  square  foot,  how  much  must  be  paid  for 
10  boards,  each  16  feet  long,  ^  foot  wide  ? 

8.  A  field  is  30  rods  long  and  24  rods  wide.  How  many 
square  rods  will  it  contain  after  a  strip  24  rods  long  and  2 
rods  wide  is  taken  from  it  for  a  road  ? 

9.  How  many  square  yards  of  plastering  will  be  re- 
quired for  a  ceiling  18  feet  long,  15  feet  wide  ? 

10.  If  a  roll  of  wall  paper  is  24  feet  long  and  18  inches 
wide,  how  many  square  yards  does  it  contain  ? 


So 

103. 

Mrs.  M.  O'Donnell. 


Chapter  One. 

BILLS. 


Chicago,  July  31,  1904. 


Bought  of  Seaver  Brothers. 


l\  yd.  Plaid 
16  yd.  Cambric 
12  pr.  Socks 

1  Wrapper 

h  yd.  Silk 

1  pr.  Gloves 

2  spools  Silk 


$1.00 
.05 
.25 

.65 

.08 


08 


1.  Copy  the  above,  filling  in  the  cost  of  each  item  and  the 
total. 

In  these  examples,  the  total  cost  of  each  item  should  be  written 
in  its  place  without  any  side  calculation.  Pupils  should  be  drilled  in 
short,  direct  methods  of  computation,  being  required  to  omit  unneces- 
sary figures. 

In  No.  2,  for  instance,  64  is  multiplied  by  5|,  as  follows: 
|  of  64  is  8  ;  carry  this  to  the  product  of  5  and  4,  making  28  ;  write 
8.     6  times  6  are  30,  add  2,  making  32.     Total,  328. 

2.  Otto  Haas  buys  of  Murphy  &  Cooper  64  pounds  of 
sugar  @  5J^ ;  28  pounds  of  lard  @  9\0;  24  pounds  of  coffee 
@25^;  1  barrel  flour  @$  5.75;  and  12  gallons  of  molasses 
@  25^.     Make  out  the  bill. 

3.  Make  out  a  bill  for  10  pairs  of  men's  shoes,  at  $  4.75 ; 
4  pairs  of  boys'  shoes,  at  $  1.47^ ;  6  pairs  slippers,  at  $  .87^ ; 
9  pairs  of  girls'  shoes,  at  $2.43;  8  pairs  of  women's  shoes, 
at  $3.37f  s 


R 


eview. 


51 


4.  Make  out  a  bill  for  8-J-  pounds  of  ham,  at  14^  per  pound ; 

3  J  pounds  of  beefsteak,  at  20^ ;  9  pounds  of  corned  beef,  at 
12^  ;  10 \  pounds  of  chicken,  at  24^ ;  12  pounds  of  roast  beef, 
at  18£ 

5.  Make  out  a  bill  for  14  dozen  collars,  at  $  1.50  per 
dozen;  6  dozen  pairs  of  culfs,  at  $2.75  per  dozen  pairs; 

4  dozen  shirts,  at  $  9  per  dozen ;  3  dozen  ties,  at  $  2.25  per 
dozen ;  17  dozen  pairs  of  socks,  at  $  2.10  per  dozen  pairs. 

104.   Eeview  Exercises.     Approximate  Answers. 

Note.  —  These  drills  are  intended  to  lead  a  pupil  to  such  an  ex- 
amination of  his  answers  to  other  problems  as  will  prevent  him  from 
being  satisfied  with  one  that  is  very  far  astray. 

It  is  not  expected  that  every  pupil  will  give  exactly  the  same 
answer.  In  No.  5,  for  instance,  the  cost  of  99  yards  is  asked  at  $  1.95 
per  yard.  One  pupil  may  consider  100  yards  at  $2,  or  $200  ;  a  sec- 
ond may  keep  the  rate  at  $1.95,  and  say  $195 ;  a  third  might  come 
still  closer ;  each  of  such  answers,  however,  should  be  accepted  as  an 
approximation. 

1.  What  will  be  the  cost  of  39f  pounds  butter  at  20^ 
per  pound  ? 

Nearly  40  pounds  at  20f.    The  cost  is  nearly  what  ?    Solve. 

2.  A  man  has  4200  pounds  of  flour  which  he  wishes  to 
put  into  barrels  containing  196  pounds  each.  About  how 
many  barrels  will  he  need  ? 

Each  barrel  contains  nearly  how  many  pounds  ?    Solve. 

3.  A  merchant  bought  a  hogshead  of  molasses,  contain- 
ing 47|  gallons,  at  50  cents  per  gallon.  About  how  much 
did  it  cost  ? 

4.  How  many  lots  at  f  1975  each  can  be  bought  for 
$12,000? 

5.  Sold  3  pieces  of  cloth,  33  yards  to  the  piece,  at  $1.95 
per  yard.     Give  the  approximate  amount  of  the  bill. 


§2  Chapter  One. 

6.  28|f  +  371$  =  nearly  what  ? 

7.  175£-j-  24T%  =  nearly  what  ? 
•     8.    18|  X  9J  =  nearly  what  ? 

9.   87TV  -  49 J|  =  nearly  what  ? 
10.   4£  x  4f  x  4^  =  nearly  what  ? 

105.   Oral  Eeview  Problems. 

1.  What  will  be  the  cost  of  8  pounds  of  meat  at  15  cents 
per  pound  ? 

2.  Gave  $  1  in  payment  for  a  25-cent  ball,  and  four  10- 
cent  bats.     How  much  change  did  I  receive  ? 

3.  At  the  rate  of  3  oranges  for  5  cents,  what  will  be  the 
cost  of  a  dozen  oranges  ? 

4.  A  gross  is  12  dozen.     How  many  pens  in  -J-  gross  ? 

5.  How  many  inches  in  4  yards  ? 

6.  At  5  cents  per  pint,  how  much  would  be  paid  for  a 
bushel  of  chestnuts  ? 

7.  A  person  used  2  gallons  and  3  quarts  of  milk  one 
week,  and  3  gallons  and  1  quart  the  next  week.  How  many 
gallons  are  used  in  the  two  weeks  ? 

8.  Multiply  15  by  5.     Take  18  from  the  product. 

9.  How  many  9's  in  3  times  21  ? 

10.  12  times  6  are  how  many  times  8  ? 

11.  To  9  times  7  add  10.     Take  15  from  the  sum. 

12.  One  can  has  in  it  4  gallons  of  milk,  and  another  has 
in  it  6  quarts.     How  many  pints  are  in  both  ? 

13.  27  +  15  +  18  +  25  +  9  =  ? 

14.  James  had  half  a  dollar  to  spend ;  he  bought  14  cents* 
worth  of  candy,  and  spent  the  rest  of  his  money  for  oranges 
at  4  cents  each.     How  many  oranges  did  he  buy  ? 


Review.  53 

15.  A  woman  bought  7  pounds  of  rice  at  12?  a  pound, 
and  paid  for  it  with  a  dollar  bill.  How  much  money  did 
she  receive  in  change? 

16.  A  man  paid  one  dollar  for  a  bag  of  peanuts  containing 
3  pecks.  He  sold  them  at  $  0.10  a  quart.  How  much  did  he 
gain? 

17.  Book,  75?;  pencil,  8^;  slate,  15?  =  ? 

18.  20  boxes  of  berries  at  15?  =  ? 

19.  At  6  cents  each,  how  many  bananas  for  $1?  How 
many  cents  over  ? 

20.  Bought  3  pounds  of  raisins  worth  12  cents  a  pound; 
2  dozen  bananas  at  25  cents  a  dozen.  I  gave  the  man  a 
dollar  bill.     How  much  did  he  give  back? 

21.  How  many  hours  are  there  in  a  week? 

22.  If  John  earned  16?  Monday,  9?  Tuesday,  20? 
Wednesday,  15?  Thursday,  8?  Friday,  and  12?  Saturday, 
how  much  did  he  earn  in  the  whole  week  ? 

23.  What  will  3  bushels  of  sand  cost,  at  4?  a  peck  ? 

24.  Mrs.  Hall  divided  84  oranges  equally  among  14  girls. 
How  many  oranges  did  each  girl  receive  ? 

25.  If  you  give  24  cents  for  one  thing,  and  19  cents  for 
another,  what  will  both  things  cost  ? 

26.  If  a  quart  of  milk  is  worth  7?,  what  is  the  value  of 
two  gallons  ? 

27.  Find  the  cost  of  60  oranges  at  20  cents  per  dozen. 

106.    Written  Eeview  Problems. 

1.  A  man  walks  14£  miles  in  4f  hours.     How   many 
miles  an  hour  is  that  ? 

2.  If  a  milk  can  holds  23  quarts  and  1  pint,  how  many 
half-pints  does  it  hold  ? 


54  Chapter  One. 

3.  Bought  87  pounds  of  tea  at  45  cents  a  pound ;  sold  it 
at  63  cents  a  pound.     How  much  was  gained  ? 

4.  In  a  school  there  were  356  girls  and  259  boys ;  if  25 
girls  and  32  boys  leave,  how  many  pupils  remain  in  the 
school  ? 

5.  Which  are  worth  more,  63  cows  at  $  38  apiece,  or  56 
horses  at  $  75  apiece  ?     How  much  more  ? 

6.  Suppose  your  mother  gave  you  a  5-dollar  bill  to  buy 
articles  for  the  Sunday  dinner,  and  you  bought  6  lb.  of  roast 
beef  at  25  cents  a  lb.,  1  pk.  spinach  at  45  cents,  2  qt.  of 
onions  at  12J  cents,  1  doz.  oranges  at  12  cents,  2  qt.  of  milk 
at  7  cents.  How  much  change  would  you  bring  home  to 
your  mother  ? 

7.  If  a  railway  mail  clerk  earns  $  800  in  a  year,  how 
much  will  he  have  left  after  paying  his  board  at  the  rate  of 
1 16  a  month  ? 

8.  How  many  pieces  of  second-class  matter  (newspapers) 
are  there  in  644  pounds,  each  piece  weighing  8  ounces  ? 

9.  The  postmaster  at  Norwich  made  requisition  for  the 
following  postage  stamps :  27  sheets  of  1-cent,  97  sheets  of 
2-cent,  35  sheets  of  5-cent,  and  17  sheets  of  10-cent  stamps. 
What  was  the  money  value  of  these  stamps,  there  being  100 
stamps  in  each  sheet  ? 

10.  The  whole  number  of  pieces  of  mail  matter  handled 
at  212  post-offices  was  2,164,517,880.  What  was  the  average 
number  of  pieces  for  each  office  ? 

11.  A  merchant  pays  $30  for  65  vases.  He  sells  17  of 
them  at  50  cents  each,  and  receives  48  cents  each  for  the 
others.     What  is  his  profit? 

12.  One  boy  had  15  marbles,  another  had  19,  a  third  had 
17,  a  fourth  had  13.  What  was  the  average  number  of 
marbles  for  each  boy  ? 


Review.  55 

13.  A  teacher  divided  200  foreign  postage  stamps  among 
the  eight  boys  of  his  class.  He  gave  one-fourth  of  them  to 
the  first  boy,  one-fifth  of  the  remainder  to  the  second  boy, 
and  then  divided  the  rest  equally  among  the  other  six  boys. 
How  many  did  each  of  the  latter  receive  ? 

14.  If  23  buggies  cost  $4025,  what  are  80  buggies 
worth  ? 

15.  How  many  gills  in  7  quarts  and  1  pint  ? 

16.  How  many  bushels  in  384  quarts? 

17.  Change  864  pints  to  gallons. 

18.  A  farmer  exchanged  16  cows,  worth  $  40  each,  for  a 
span  of  horses.     What  are  the  horses  worth  apiece  ? 

19.  A  boy  bought  a  bicycle  for  $35.  He  rented  it  to 
another  boy  for  3  months  at  $  2  a  month,  and  then  sold  it 
for  $33.50.     Did  he  gain  or  lose,  and  how  much  ? 

20.  John  had  16  marbles,  Henry  half  as  many,  and 
Frank  as  many  as  both  the  other  boys.  How  many  more 
marbles  had  Frank  than  John  ? 

21.  How  many  quarts  in  12  bushels  ? 

22.  How  many  feet  of  string  will  be  required  to  go 
around  a  room  30  feet  long  and  25  feet  wide  ? 

23.  If  I  buy  a  bushel  of  walnuts  for  $3,  and  sell  them  at 
5  cents  a  pint,  how  much  do  I  make  ? 

24.  Write  83,  47,  69,  and  56  in  Eoman  numbers. 

25.  A  man  works  9  months,  26  days  per  month,  and 
receives  $  702.     What  are  his  daily  wages  ? 

26.  A  merchant  buys  136  vases  for  $272.  If  36  are 
broken,  how  much  must  he  charge  apiece  for  the  others  to 
gain  $28  on  all? 


56  Chapter  One. 

27.  On  Monday,  the  receipts  of  a  store  are  $180;  on 
Tuesday,  $30  less ;  on  Wednesday,  $  110  less  than  the  total 
of  Monday  and  Tuesday.  What  are  the  receipts  for  the 
three  days  ? 

28.  The  yearly  rent  of  a  house  is  $480.  What  is  the 
rent  for  2  years  4  months  ? 

29.  A  mechanic  works  300  days  per  year,  at  $  4  per  day. 
If  his  daily  expenses  for  365  days  average  $  3,  how  much 
money  does  he  save  each  year  ? 

30.  A  woman  pays  $5.20  for  3  pounds  of  tea  and  56 
pounds  of  sugar.  What  is  the  price  per  pound  of  the  sugar, 
if  the  tea  costs  80^  per  pound  ? 

31.  A  man  had  $7500.  He  paid  -J  of  it  for  a  house, 
$  575.60  for  repairs,  and  $  387.75  for  furniture.  How  much 
money  had  he  left  ? 

32.  How  much  hay  will  be  required  to  feed  18  horses 
a  month  of  30  days,  if  each  horse  receives  15  pounds. a  day? 

33.  A  person  pays  a  debt  of  $576,  giving  40  ten-dollar 
bills,  30  two-dollar  bills,  6  one-dollar  bills,  and  the  re- 
mainder in  five-dollar  bills.  How  many  of  the  last  did 
he  give  ? 

34.  A  drover  buys  64  sheep  for  $400.  He  sells  -$-  of 
them  at  $7  each,  and  the  remainder  at  $8  each.  What 
is  his  profit? 

35.  A  merchant  sells  56  yards  of  cloth  for  $  84,  gaining 
$  14.     What  did  it  cost  him  per  yard  ? 

36.  A  package  of  coffee,  costing  60  cents,  was  sold  for  75 
cents,  the  profit  on  each  pound  being  5  cents.  What  was 
the  selling  price  per  pound  ? 

37.  How  many  yards  of  cloth,  at  $1.75  per  yard,  can  be 
bought  for  $105? 


Review.  57 

38.  A  tailor  buys  a  piece  of  cloth  for  $50.  From  it 
he  makes  4  pairs  of  trousers,  which  he  sells  at  $  7  per  pair, 
and  4  coats,  for  each  of  which  he  receives  $15.  Thread, 
buttons,  lining,  etc.,  cost  him  $  16.  How  much  does  he  get 
for  his  labor  ? 

39.  A  man  sold  a  certain  number  of  papers  for  50  cents. 
If  he  had  sold  nine  more,  he  would  have  received  95  cents. 
How  many  papers  did  he  sell  ? 

40.  How  long  is  a  post  which  is  5|-  feet  above  water, 
one-half  of  its  length  in  the  water,  and  one-fourth  of  its 
length  in  the  mud  ?     (Make  a  diagram.) 

41.  Eight  pounds  of  black  tea  costing  35^  per  pound  are 
mixed  with  twelve  pounds  of  green  tea  costing  50^  per 
pound.     What  is  the  cost  of  20  pounds  of  the  mixed  tea  ? 

42.  How  many  bushels  and  pecks  are  there  in  1442 
pounds  of  corn  weighing  56  pounds  per  bushel? 

43.  How  is  division  proved  ? 

44.  Multiply  by  208  the  quotient  of  (169,668  -j-  36). 

45.  Add  seventy-two  dollars,  eleven  cents;  fifteen. dollars, 
nine  cents;  eighty-seven  cents;  three  hundred  fifty  dollars; 
and  one  dollar,  four  cents. 

46.  Which  is  greater  and  how  much  ? 

486  x  29  or  26,845  - 19,976. 

47.  Write  in  Eoman  numerals  1905,  1775,  and  560. 

48.  If  a  railway  mail  clerk  spends  ten  cents  a  day  for 
street-car  fare,  how  much  will  he  spend  in  six  months  of  30 
days  each? 

49.  Add  nine  thousand  eleven,  seventy  thousand  forty- 
four,  five  hundred  thousand  four  hundred  ten,  fifty-four 
thousand  twenty-one. 

50.  Multiply  $40.25  by  96. 


58  Chapter  One. 

51.  From  $300,000  take  $7050.75. 

52.  How  many  days  will  36  bushels  of  oats  last  12 
horses,  if  each  horse  eats  12  quarts  a  day  ? 

53.  If  a  barrel  of  flour  is  worth  $4.50,  how  many  barrels 
can  be  bought  for  $441?  How  much  will  all  the  flour 
weigh  if  each  barrel  holds  196  pounds? 

54.  Suppose  your  slate  is  12  inches  long,  9  inches  wide, 
and  15  inches  across  diagonally.  How  long  a  string  is 
needed  to  go  around  the  outside  and  along  the  diagonal  ? 
Make  a  diagram  to  explain  your  work. 

55.  The  total  cost  of  the  Union  Pacific  railroad,  which 
is  1819  miles  long,  was  $157,092,478.  What  was  the  aver- 
age cost  per  mile  ? 

56.  An  officer  who  was  paid  $3.50  a  day  stayed  in  the 
service  until  he  had  earned  $  143.50.  How  many  days  had 
he  worked? 

57.  A  cargo  of  potatoes  was  discharged  in  tubs  contain- 
ing 250  pounds  each,  which  were  filled  1785  times.  A 
bushel  of  potatoes  weighs  60  pounds.  How  many  bushels 
were  landed  ? 

58.  How  long  will  it  take  50  clerks  to  count  $1,500,000 
in  silver  coin,  one-half  of  which  is  in  half-dollars  and  the 
other  half  in  quarter-dollars,  each  clerk  counting  at  the  rate 
of  fifty  pieces  a  minute  ?    Express  the  answer  in  hours. 

59.  Write  in  figures  one  million  one  thousand  one  dollars 
and  one  cent. 

60.  Multiply  657,934  by  3209. 

61.  The  War  Department  expended  $  1765.25  for  muci- 
lage at  $5.75  a  dozen  quarts.  How  many  quarts  were  pur- 
chased ? 


CHAPTEE  II. 

PAGES 

Fractions 69  to  84 

Greatest  Common  Divisor,  Least  Common  Multiple, 
Addition  and  Subtraction  of  Fractions,  Cancellation, 
Ratio,  Multiplication  and  Division  of  Fractions. 

Decimals 84  to  91 

Multiplication  of  Decimals,  Division  of  Decimals. 

United  States  Money 92  to  93 

Fractional  Farts  of  a  Dollar. 

Denominate  Numbers 94  to  99 

Reduction,  Addition,  Subtraction,  Multiplication,  and 
Division. 

Measurements 99  to  101 

Areas  and  Surfaces. 

Bills .        .        .  101  to  102 

Review  of  Simple  Numbers 102  to  118 

Short  Methods,  Sight  Exercises,  Sight  Approximations, 
Review  Problems. 

FACTORS   AND  MULTIPLES. 

107.  The  factors   of   a  number   are  the   integers  whose 
product  makes  the  number. 

Note.  —  An  integer  is  any  whole  number. 

2  and  3  are  factors  of  6. 
2,  3,  and  5  are  factors  of  30. 

108.  A   number  that  contains   another   number    an    exact 
number  of  times  is  a  multiple  of  that  number. 

24  is  a  multiple  of  12;  36, 48,  etc.,  are  also  multiples  of  12. 
30  is  a  multiple  of  2,  3,  5,  6,  10,  15. 

69 


60  Chapter  Two. 

109.  Preliminary  Exercises. 

1.  95  is  a  multiple  of  what  two  numbers  ? 

2.  Give  the  two  factors  of  51. 

3.  What  number  is  a  multiple  of  both  8  and  6  ? 

4.  Mention  another  number  that  is  a  multiple  of  both  8 
and  6. 

5.  Find  the  smallest  number  that  can  be  exactly  divided 
by  8  and  12. 

6.  Give  the  two  factors  of  91. 

7.  57  is  a  multiple  of  what  two  numbers  ? 

8.  What  is  the  smallest  number  that  can  be  exactly 
divided  by  4,  6,  and  8  ? 

PRIME   NUMBERS. 

110.  A  number  that  has  no  factors  is  a  prime  number* 

Note.  — 1  is  not  considered  a  factor. 

1.  2,  3,  5,  7,  etc.,  are  prime  numbers. 

111.  1.   Name  the  prime  numbers  between  10  and  20. 

2.  Between  20  and  30.  4.   Between  50  and  70. 

3.  Between  30  and  50.  5.   Between  70  and  100. 

112.  Oral  Exercises. 

Give  the  prime  factors,  commencing  with  the  smallest. 

1.  15  6.   40  11.   64  16.   80 

2.  16  7.   48  12.    72  17.    81 

3.  24  8.    54  13.    74  18.    82 

4.  32  9.   56  14.    76  19.   84 

5.  36  10.   60  15.   77  20.   85 


by 

ne 

2 
3 
3 

90 

45 

15 

5 

17. 

576 

18. 

840 

19. 

1152 

20. 

1728 

21. 

2016 

Fractions.  6i 

113.  Written  Exercises. 

1.  Find  the  prime  factors  of  180. 

Divide  180  by  its  smallest  prime  factor,  2.    Divide  the        2|180 
quotient  90  by  its  smallest  prime  factor,  2.     Divide  45  by- 
its  smallest  prime  factor,  3.     Divide  15  by  its  smallest  prime 
factor,  3.     The  quotient  5  is  a  prime  number. 

The  prime  factors  of  180  are  2,  2,  3,  3,  5,  Ans. 

2.  86  7.  92  12.  100 

3.  87  8.  93  13.  120 

4.  88  9.  94  14.  210 

5.  90  10.  95  15.  240 

6.  91  11.  96  16.  360 

GREATEST  COMMON  DIVISOR. 

114.  A  common  factor  of  two  or  more  numbers  is  a  num- 
ber that  wijl  divide  each  of  them  without  remainder. 

The  largest  number  that  is  a  factor  of  two  or  more  numbers 
is  called  the  greatest  common  divisor. 

115.  Oral  Exercises. 

Find  the  greatest  common  divisor  of: 

1.  27  and  48  6.  34  and  51 

2.  25  and  35  7.  32  and  48 

3.  36  and  54  8.  45  and  75 

4.  26  and  39  9.  40  and  65 

5.  42  and  63  10.  54  and  69 


62  Chapter  Two. 

LOWEST  TERMS. 

116.  How  can  you  tell  that  a  number  is  divisible  by  2  ? 
By  5? 

A  number  is  divisible  by  3  when  the  sum  of  its  digits 
(figures)  is  divisible  by  3 ;  it  is  divisible  by  9  when  the  sum 
of  its  digits  is  divisible  by  9. 

A  number  is  divisible  by  4  when  the  number  expressed 
by  its  last  two  figures  is  divisible  by  4. 

When  is  a  number  divisible  by  25  ? 

A  fraction  is  reduced  to  lowest  terms  by  dividing  the 
numerator  and  the  denominator  by  their  greatest  common 
divisor. 

117.  "Written  Exercises. 

1.  Reduce  £f§  to  its  lowest  terms. 

A  look  at  both  terms  shows  that  3  is  a  common  factor.  This 
reduces  the  fraction  to  fife.  41  is  a  prime  number,  and  is  not  a  factor 
of  100,  so  that  T^y  cannot  be  reduced  to  lower  terms. 

2.  Reduce  Jj-f-  to  its  lowest  terms. 

4  +  3  +  2  =  9;    6  +  2  +  1=9. 
Since  the  sum  of  the  digits  of  each  term  is  divisible  by  9,  this  num- 
ber is  a  common  factor,  and  reduces  the  fraction  to  £$,  etc. 

3.  Reduce  £§f  to  its  lowest  terms. 
5  is  clearly  a  common  factor,  etc. 

Reduce  to  lowest  terms : 

*•  m  *■  m  "•  ftt 

«•  m  »•  m  is-  m 

6-   H(  io.  ft  14.   |H 

7.  w  ii.  u  i5-  m 

118.  Reduce  to  its  lowest  terms  TVfr 

In  this  example,  it  is  not  easy  to  ascertain  by  inspection  any  num- 
ber that  will  divide  both  terms.    In  such  cases,  the  greatest  common 


Fractions.  63 


divisor  is  found  by  dividing  the  denominator  by  the  numerator.  The 
remainder  is  divided  into  the  numerator,  and  each  subsequent  remainder 
is  divided  into  the  corresponding  divisor  until  there  is  no  longer  a 
remainder.  This  last  divisor  is  the  greatest  common  divisor  of  the  two 
numbers. 

The  numerator,  169,  is  contained  in  the  5 

denominator,   1001,    5   times    with    156   re-        169)1001 
mainder.    This    remainder    is    contained  in  845       1 

the  numerator,    169,  one   time  with  13  re-  156)169 

mainder.     This  remainder   is    contained  in  156     12 

the  previous  divisor,  156,  12  times  with  no  13)156 

remainder.  13 

13  is  the  greatest  common  divisor.  ~Kq 

169  +  13      13  ,  26 

=  ==  lowest  terms.  — 


1001  -=-  13      77 

Note.  —  In  reducing  fractions  to  lowest  terms,  the  method  of 
finding  the  greatest  common  divisor  given  above  should  not  be  resorted 
to  if  it  is  possible  to  get  along  without  it. 


119.   Written  Exercises. 

Keduce  to  lowest  terms : 

i-  » 

5. 

m 

9. 

u 

2-    Hi 

6. 

22T 

10. 

TFT 

3.  « 

7. 

frV 

11. 

ttf 

4-    II 

8. 

t¥* 

12. 

m 

Suggestion. — Do  not  waste  time  in  finding  the  greatest  common 
divisor. 

13.  ^  17.  {fa  21.  Jy& 

14.  #V  18.  £k  22.  fitffo 

15.  t^  19.  A%  23.  T^fo 
16-  TWir  20-  libs  24.  TU 


64  Chapter  Two. 

LEAST  COMMON  MULTIPLE. 

120.  The  smallest  number  that  is  a  multiple  of  two  or  more 
numbers  is  called  the  least  common  multiple  of  such  numbers. 

121.  Oral  Exercises. 

Give  the  least  common  multiple  of: 


1. 

16  and  24 

6. 

2,  3,  5,  9,  10 

2. 

12  and  15 

7. 

2,  3,  5,  6,  9,  10 

3. 

3,  9,  11 

8. 

3,  6,  9,  12,  4,  18 

4. 

4,  12,  16 

9. 

2,  7,  14,  3,  9 

5. 

2,  3,  4,  5,  6 

10. 

5,  10,  20,  25,  50 

2  3  9  J  U    0  14  2  12 

3 

9          7     6 

122.  Written  Exercises. 

Find  the  least  common  multiple  of  3,  9,  7,  14,  6,  14,  2,  12. 

3  is  stricken  out  since  it  is  a 
factor  of  6,  which  is  one  of  the 
numbers.     7  is  a  factor  of  14,  3  7  2 

one  14  is  stricken  out.     6  is  a 

factor  of  12.     2  is  a  factor  of  12.    The  least  common  multiple  of  the 
remaining  numbers,  9,  14,  and  12,  is  to  be  found. 

Divide  these  numbers  by  a  prime  number  that  is  exactly  contained 
in  any  two  of  them,  bringing  down  the  numbers  that  are  not  multiples 
of  the  divisor. 

Taking  2  as  a  divisor,  bring  down  9,  and  write  quotients  7  and  6. 

3  being  a  factor  of  two  of  the  three  numbers,  9,  7,  6,  is  taken  as  the 
next  divisor.  3  is  written  as  a  quotient,  7  is  brought  down,  2  is  a 
quotient. 

As  there  is  no  factor  common  to  any  two  of  the  numbers,  3,  7,  2, 
we  find  the  least  common  multiple  by  multiplying  together  the  two 
divisors  and  these  three  numbers. 

2  x  3  x  3  x  7  x  2  =  252  L.  C.  M. 

123.  Find  the  L.  CM.  of: 

1.   4,  6,  3,  5,  8,  20  2.   9,  15,  15,  4,  4,  12,  25 

Strike  out  4,  3,  5.  Strike  out  one  16  and  two  4's. 


Fractions. 


65 


3.  2,  3,  5,  7,  5,  14,  10,  12,  24 

4.  2,  3,  5,  6,  8,  10,  15,  16,  80 

5.  20,  30,  40,  50 

6.  2,  3,  4,  6,  8,  12,  16,  24 

7.  24,  12,  5,  3,  10,  18  9.    18,  5,  9,  40,  16 

8.  11,  3,  7,  77,  33  10.    10,  12,  15,  21 

ADDITION  AND   SUBTRACTION  OF  FRACTIONS. 

124.  In  adding  or  subtracting  fractions,  they  must  be  reduced 
to  a  common  denominator. 

The  least  common  denominator  is  the  least  common  multiple 
of  the  denominators. 


125.   Add  the  fractions,  },  ft,  f,  ft,  f  f,  &. 
£    20     0    30    45     12 


10 


U    45      6 


45      £ 
L.  C.  M.  =  2  x  2  x  45  =  180 


Strike  out  4  and  6. 

Strike  out  15,  a  factor  of  45. 

Strike  out  5  and  3,  factors  of  45. 


180 

f 

135 

H 

99 

1 

150 

« 

102 

H 

92 

& 

105 

Ans.   8Hf 

H*  = 

3H§- 

To  add,  reduce  the  fractions  to  a  common  denominator,  add 
the  numerators,  and  place  the  sum  over  the  common  denom- 
inator.   Reduce  if  possible. 

To  subtract,  reduce  the  fractions  to  a  common  denominator? 
subtract  the  numerators,  and  place  the  difference  over  the 
common  denominator.     Reduce  if  possible. 


66  Chapter  Two. 

Note In  the  following  examples,  determine  the  least  common 

denominator  by  inspection,  if  possible. 

126.  Add: 

1.  8f,  5},  3J  6.  |,  f,  A,  A,  H 

2.  45f,  20|,  8f,  9£  7.   63^,  3|,  24,  5f,  7^ 

3.  32},  19f i  6J,  8£|  8.   5&,  18^,  34,  7|,  8} 

4.  2£,  20,  3f ,  ^,  54.  9.   4^,7^,84,6^,^ 

5.  84,  454 ,  2J4,  4J,  |4     10.   17^  rffo  6&,  ^  ^ 
11.  Work  No.  9  as  an  example  in  decimals. 

12-   Work  No.  10  as  an  example  in  decimals. 

127.  Subtract: 

13.  65H-   57^  18.   251^-   -   27^ 

14.  18tf-     9f|  19.   755£f   -283ff 

15.  100||-    15|f  20.   123£}   -   80£f 

16.  102^-    27«  21.    100^-   89^ 

17.  208^-1284$  22.     67^   -    58^ 

23.  Work  No.  21  as  an  example  in  decimals. 

24.  Work  No.  22  as  an  example  in  decimals. 

128.  Perform  the  operations  indicated : 

OK     21  +  5     21 


9A 

25  +  5     25 
21     21-5 

<0O. 

25     25-5 

27. 

(37|-llf)- 

-(28rV 

-19A> 

28. 

14f-8i-3f  +  4J 

29. 

(8ft  +  6|)- 

•(»A- 

<® 

Fractions.  67 

30.  4f  x  16  x  8£ 

31.  (M**k}Mm-H+m 

32.  (8i  +  4J)-(2i  +  li) 

33.  (3^x36)x8f 

34.  4|  +  3i-6f  +  17i-9f 

129.    Oral  Problems. 

1.  A  person  travelling  from  New  York  to  Albany  (140 
miles  apart)  has  gone  102J  miles.  How  much  farther  has 
he  to  go  ? 

2.  There  are  196  pounds  of  flour  in  a  barrel.  How 
many  pounds  in  J  barrel  ? 

3.  How  many  fourths  in  24 J  ? 

4.  Eeduce  f$  to  lowest  terms. 

5.  Change  ±$Q-  to  a  mixed  number. 

6.  Add  i,  ^,  and  J. 

7.  From  a  chest  of  tea  containing  45 J  pounds,  14| 
pounds  are  sold.     How  many  pounds  remain  ? 

8.  From  ^  of  a  dollar  take  10|  cents. 

9.  How  many  cents  in  ^  +  ^  -f-  y%  of  a  dollar? 

10.  A  farmer  has  60|  acres  of  pasture  and  20|  acres  of 
woodland.     How  many  acres  in  both  ? 

11.  Considering  the  circumference  of  a  circle  as  3^  times 
its  diameter,  find  the  circumference  of  a  circle  whose  diam- 
eter is  8  feet. 

12.  Mary  is  12^  years  old;  Jane  is  3fJ-  years  older* 
How  old  is  Jane  in  years  and  months  ? 

13.  In  a  year  and  a  half  William  will  be  7  years  2  months 
old.     How  old  is  he  now  ? 


68  Chapter  Two. 

14.  What  number  multiplied  by  3  equals  231  ? 

15.  What  number  between  7  and  12  is  a  prime  number  ? 

16.  A  boy  received  9  marks  in  arithmetic,  8  in  penman- 
ship, and  7  in  reading.     What  was  his  average  mark  ? 

17.  -f  of  a  class  consists  of  boys.  How  many  girls  in  the 
class,  if  it  contains  49  pupils  ? 

18.  When  July  1  falls  upon  Tuesday,  what  will  be  the 
date  of  the  third  Tuesday  of  July  ? 

19.  If  July  1  falls  upon  Thursday,  upon  what  day  will 
the  first  of  August  fall  ? 

20.  A  man  bought  20|-  pounds  of  sugar;  he  sold  10| 
pounds  at  one  time  and  6J  pounds  at  another.  How  much 
had  he  left  ? 

21.  If  3  quarts  1  pint  of  oil  cost  7  cents,  what  will  1  gal- 
lon 1  quart  cost  ? 

22.  How  much  will  have  to  be  paid  for  7  cows  at  $50 
each,  and  4  horses  at  $  150  each  ? 

23.  f  =  how  many  hundredths? 

24.  What  are  the  two  factors  of  87  ? 

25.  Find  the  G.  C.  D.  of  36  and  54. 

26.  If  eggs  are  sold  at  the  rate  of  21  for  25  cents,  how 
much  will  be  paid  for  5  \  dozen  ? 

Suggestion. — Every  member  of  the  class  should  be  required  to 
solve  one  of  the  foregoing  examples  as  a  sight  problem,  first  reading  it 
from  the  book,  and  then  giving  the  answer.  No  time  should  be  wasted 
in  "analyzing"  the  problems,  unless  some  pupil  desires  the  explana- 
tion of  one  that  he  does  not  understand. 

At  another  time,  the  teacher  should  read,  say,  five  or  ten  problems, 
requiring  the  answer  to  each  to  be  written,  at  a  given  signal,  and  the 
pencil  laid  down  before  the  next  is  read.  No  alteration  of  an  answer 
should  be  permitted. 


Fractions.  69 

130.   Written  Problems. 

1.  A  horse  travelled  48^-  miles  in  one  day,  56|  the 
next,  40i|  the  third,  and  45f-J  the  fourth.  How  far  did  he 
travel  in  all  ? 

2.  To  the  sum  of  6£  and  19|  add  their  difference. 

3.  From  a  bin  containing  25f  bushels  of  grain  there 
were  taken  out  5f  bushels  at  one  time  and  6J  at  another. 
How  much  remained  ? 

4.  A  merchant  sold  4  pieces  of  cloth  containing  21\ 
yards,  26f  yards,  29f  yards,  and  28±-  yards,  respectively. 
How  much  did  he  receive  for  the  cloth  at  96  cents  per  yard  ? 

5.  Keduce  ||-|  to  lowest  terms. 

6.  A  man  has  8^-  bushels  of  peanuts.  He  puts  them 
into  bags  holding  -^  bushel.     How  many  bags  does  he  fill  ? 

7.  A  160-acre  farm  consists  of  five  fields ;  the  first 
contains  17-f  acres,  the  second  29£  acres,  the  third  35^j- 
acres,  the  fourth  22-fe  acres.  How  many  acres  are  there  in 
the  fifth  field  ? 

8.  Prom  a  piece  of  silk  that  contained  28£  yards,  there 
were  sold  2\  yards,  6£  yards,  and  13|  yards.  Find  the  value 
of  the  remainder  at  $1.20  per  yard. 

9.  Three  pieces  of  cloth  bought  at  $2  per  yard  cost 
$150.  The  first  piece  measures  23£  yards,  the  second 
measures  30f  yards.     How  many  yards  in  the  third  piece  ? 

10.  What  part  of  a  person's  income  remains  after  he 
spends  J,  -^,  and  \  of  it? 

11.  A  boy  loses  \  of  his  marbles,  and  he  gives  away  \  of 
them.     If  he  has  17  marbles  left,  how  many  had  he  at  first  ? 

12.  A  dealer  sells  If  gross,  3J  gross,  and  8|  gross  of  lead 

pencils  at  36  cents  per  dozen.     How  much  does  he  receive 

for  all  ? 

1  gross  =  12  dozen. 


70  Chapter  Two. 

13.  There  are  four  towns,  A,  B,  C,  and  D,  on  a  certain 
railroad  running  east  and  west.  A  is  41^-  miles  west  of  C ; 
D  is  39J  miles  east  of  B ;  B  is  22£  miles  west  of  C.  How 
many  miles  from  A  to  D  ?     Make  a  diagram. 

CANCELLATION. 

131.  Preliminary  Exercises. 

1.  Divide  64  by  16.     The  quotient  is  4. 

2.  Divide  \  of  64  by  £  of  16 ;  i.e.  32  -*-  8. 

3.  Divide  \  of  64  by  J  of  16 ;  i.e.  16  -f-  4. 

4.  Divide  \  of  64  by  £  of  16 ;  i.e.  8  -h  2. 
In  each  case  the  quotient  is  4. 

In  example  2  we  took  out  of  the  dividend  64  the  factor  2,  making 
the  new  dividend  32  ;  and  we  took  out  of  16  the  same  factor,  making 
the  new  divisor  8. 

In  example  3  we  took  what  factor  out  of  the  divisor  and  the  divi- 
dend ?    What  common  factor  was  taken  out  in  example  4? 

Rejecting  the  same  factor  from  the  divisor  and  the  dividend 
does  not  change  the  quotient. 

In  reducing  -fj  to  J  what  factor  common  to  the  numerator 
and  the  denominator  of  the  first  fraction  is  rejected  ?  Is 
the  value  of  the  first  fraction  altered  by  this  rejection? 

Cancellation  is  the  striking  out  of  common  factors  from  the 
divisor  and  the  dividend. 

132.  Oral  Exercises. 

36  x!4  42  x23  67  x  36  83  x  36 

9  21  18  '    '    12 

2-fx16    6-2^x46    10-ix82     14-ifx48 

3'12xi       7-32xI        "'     4Xi        15'15X| 
25x18  33x12  89  x  13  44x17 

36  ■   '    99  26  34 


Fractions.  71 

RATIO. 

133.  Preliminary  Exercises. 

1.  If  oranges  are  worth  28  cents  a  dozen,  what  will  be 
the  cost  of  3  oranges  ? 

2.  What  part  of  a  dozen  is  3  ? 

3.  What  is  the  ratio  of  3  to  12  ? 

Ratio  is  the  relation  between  two  like  numbers.     It  is  found 
by  dividing  the  first  by  the  second. 

4.  What  is  the  ratio  of  12  to  16  ? 

5.  If  16  apples  cost  a  certain  sum,  what  part  of  this  sum 
should  be  paid  for  a  dozen  apples  ? 

134.  Written  Exercise. 

1.  If  17  horses  cost  $4000,  what  will  be  the  cost  of  51 
horses  at  the  same  price  for  each  ? 

Since  the  ratio  between  51  and  17  is  f^  or  3,  51  horses  will  cost 
3  times  §4000,  or  $12,000. 

2.  If  15  eggs  cost  25  cents,  what  will  10  dozen  cost? 

10  x  12 
The  ratio  of  10  dozen  eggs  to  15  eggs  is  — - — • 

15 

Multiply  25  cents  by  -10  x  12. 
15 

In  this  case,  15  is  not  contained  in  any  number 

above  the  line.     We  divide  15  and  10  by  5,  cancel-  2         4 

ing  them  and  writing  quotients  3  and  2  alongside.  25  X  %$  X  \% 

3  is  contained  in  12  4  times  ;  so  we  cancel  3  and  12.  J.fi 

Our  answer  now  is  25  cents  x  2  x  4  =  200  cents,  p 

or  $2. 

3.  Eighteen  men  can  do   a  piece  of  work  in  26  days. 
How  long  will  it  take  13  men  to  do  the  same  work? 

Thirteen  men  will  do  the  work  in  \\  of  the  time  required  by  18 
men. 


72  Chapter  Two. 

4.  Seventeen  barrels  of  flour,  196  pounds  each,  were  put 
into  bags  holding  49  pounds  each.  How  many  bags  of 
flour  were  put  up? 

5.  At  the  rate  of  23  cents  for  7  pounds,  how  much  would 
be  paid  for  91  pounds  of  flour? 

6.  A  bank  pays  $  4  interest  a  year  on  every  $  100.  How 
much  interest  will  be  paid  for  3  years  on  $650? 

7.  At  $  7.50  per  thousand  for  bricks,  what  must  I  pay  for 
63,200  bricks? 

8.  If  flour  is  $  6  per  barrel  (196  lb.),  what  must  be  paid 
for  a  49-pound  bag? 

9.  A  grocer  buys  84  dozen  eggs,  and  sells  them  at  the 
rate  of  21  eggs  for  25  cents.  What  does  he  receive  for 
them? 

10.  A  miller  buys  9840  pounds  of  wheat  at  90  cents  per 
bushel  of  60  pounds.     How  much  does  he  pay  for  it  ? 

11.  What  will  be  the  cost  of  64  sheep,  if  18  cost  $  198  ? 

12.  If  18  men  can  do  a  piece  of  work  in  42  days,  how 
long  will  it  take  21  men  to  do  the  same  work? 

13.  What  will  be  the  cost  of  66  dozen  pens  at  42  cents 
per  gross  of  12  dozen? 

14.  A  certain  quantity  of  hay  feeds  15  horses  56  days. 
How  long  will  it  feed  14  horses  ? 

15.  A  merchant  bought  9  pieces  of  cloth,  each  containing 
24  yards,  for  $  189.     What  was  the  price  per  yard  ? 

MULTIPLICATION  OF  FRACTIONS 

135.   Preliminary  Exercises. 

What  is  i  of  2  fifths  ?     Of  4  sevenths  ?    Of  6  elevenths  ? 

What  is  \  of  i?    Of£?    Of  J?    Of  J?    Show  by  diagram0 

What  is  \  of  |?     Off?    Off?    Off? 

What  is  \  of  f?    fof£?    fof£? 


Fractions.  73 

What  is  -§-  of  I?     I  off?    foff? 

What  is  -I  of  i?    ioff?    fofj?    foff? 

What  is  the  half  of  1£?    0f2i?     0f3|?     Of  4^? 

What  is  one-third  of  1£  ?     J  of  1\  ?    £  of  2\  ?    f  of  2J  ? 

136.  "Written  Exercises. 

1.  Multiply  £  by  f. 
This  means  to  find  #  of  f . 

Since  *  of  *  =  A»  i  of  *  =  A.  and  }  of  f  =  &,  or  |  x  f  =  &. 

One  fraction  is  multiplied  by  another  by  placing  the  product 
of  the  numerators  over  the  product  of  the  denominators  in  the 
form  of  a  fraction. 

Note.  —  Cancel  when  possible. 

2.  Multiply  I  by  A- 

i°f  A  =  A  £<>f  A  =  2timesA  =  f 

3 

Cancel  2  and  10,  writing  5  under  10.     Cancel  %  ^  ft  _  3 

3  and  9,  writing  3  above  9.  £      20      5 


Show  by  a  diagram  that  2  times  ^  is  \. 


5 


3.  Multiply  12i  by  3^. 
Reduce  the  mixed  numbers  to  improper  fractions. 

17      7 

g2     40  =  119== 

3 

4.  Multiply  117  by  3|. 

The   multiplication  of  an  integer  by  a  mixed 
number,  or  of  a  mixed  number  by  an  integer,  can        13 
be  considered  as  multiplication  of  fractions,  the       2ML  v  _  —  377 
integer  being  written  as  an  improper  fraction  with         1        ^ 
1  for  the  denominator. 


74  Chapter  Two. 

137.   Multiply: 

1.  |  by  96  16.   &x$% 

2.  128  by  |  17.   3fbyl2£ 

3.  *  by  |  18.   $  x  4« 


ibyf  19.  fbyfbyH 

5.  *  by*  20.  fj  of  |f  of  4. 

6.  3^  by  72  21.  |}  x  f  X  * 

7.  24fbyl8  22.  ft  of  ft  of  ff 


8.  69fby32  23.  J  of  65f 

9 
10 

11.    2£by3f  26.   4^x5^ 

12 
13 


111^  by  28  24.   fof55f 

67by^  25.    6Jx7| 


14. 
15. 


AX2J-  27.  §  of  4^x3* 

17iby6f  28.  |of3|x4IV 

6*xf  29.  Hx2Jx3J 

4Jby8f  30.  2£x2Jx2J 


138.   Perform  the  indicated  operations : 
Note.  — \  of  3 J  is  the  same  as  \  x  3 J,  or  3 J  x  $. 


1. 

J  of  (3H-6J) 

6. 

(8}  x  21) -(*  of  15© 

2. 

<3*-2»X| 

7. 

5i  +  6J  +  7J 

3. 

iof(5i-3f) 

8. 

18f-8|-7J 

4. 

(24J  +  16i)H-8 

9. 

f  off  of  (3*  + If) 

6. 

(3i+2*)x(3i- 

•2fi 

10. 

(l8i-6f) +11 

139.    Oral  Exercises. 

1.  Sold  a  house  lot  for  $30,  which  was  J  of  what  it 
cost  me.     What  was  the  cost  of  the  lot  ? 

2.  A  man  can  mow  6  J  acres  of  grass  in  a  day.     How 
much  can  he  mow  in  6  days  ? 


Fractions.  75 

3.  A  man  bought  15  bushels  of  corn  for  1\  dollars. 
How  much  did  a  bushel  cost  ? 

4.  A  boy  is  18  years  old  and  his  age  is  f  of  the  age  of 
his  father.     How  old  is  his  father  ? 

5.  Cloth  is  worth  fo  of  a  dollar  a  yard.  What  is  f  of 
a  yard  worth  ? 

6.  At  the  rate  of  5  cents  for  \  of  a  pie,  for  how  many 
pies  will  a  man  receive  $  1.60  ? 

7.  What  would  |-  of  a  yard  of  carpet  cost  at  f  of  a 
dollar  a  yard  ? 

8.  I  had  ^2  °f  a  pound  of  candy  and  gave  away  f  of  it. 
What  part  of  a  pound  did  I  give  away  ? 

9.  What  will  15  yards  of  ribbon  cost  at  6§  cents  a  yard  ? 

10.   What  will  2f  gallons  of  ice-cream  cost  at  If  dollars 
a  gallon  ? 

140.   "Written  Exercises. 

1.  A  man  worked  6  days  at  2}  dollars  per  day,  his  son 
5  days  at  If  dollars,  his  daughter  4  days  at  £  of  a  dollar. 
What  were  their  total  earnings  ? 

2.  A  merchant  bought  a  piece  of  cloth  for  28|  dollars 
and  was  obliged  to  sell  it  for  -|  of  what  it  cost  him.  How 
much  did  he  lose  ? 

3.  A  hotel  in  one  month  used  31  pounds  of  coffee  and 
7f  times  as  much  sugar.     How  much  sugar  was  used  ? 

4.  A  man  gave  124T5g-  acres  of  land  to  his  two  sons, 
giving  f  of  it  to  the  elder  and  ■§■  to  the  younger.  How 
many  acres  did  each  receive? 

5.  If  it  requires  21|  days  for  a  man  to  dig  a  ditch, 
what  part  can  he  dig  in  15  days  ? 


7 6  Chapter  Two. 

6.  If  a  bird  can  fly  lOf  miles  in  f  of  an  hour,  how  far 
can  it  fly  in  2£  hours  ? 

7.  What  would  be  the  cost  of  a  side  of  veal  containing 
52  pounds  at  9 \  cents  a  pound  ? 

8.  What  will  16  pairs  of  shoes  cost  at  %  3}  a  pair  ? 

9.  A  man  who  owed  $  7825  failed  and  could  pay  only 
|  of  his  debts.     How  much  could  he  pay  ? 

10.  I  bought  a  house  and  lot  and  made  a  payment  of 
%  4500,  which  was  f  of  the  cost.  What  was  the  cost  of  the 
property  ? 

DIVISION  OF  FRACTIONS. 
141.   Preliminary  Exercises. 

1.  If  3  yards  of  calico  cost  18  cents,  what  is  the  price 
per  yard  ? 

18  j*  -T-  3,  or  \  of  18/?.     The  latter  may  be  written  18  j*  x  \. 

2.  If  \\  yards  of  dress  goods  cost  18^,  what  is  the  price 

Per  yard?  i8^H,  or  18^|. 

To  divide  18  by  f ,  we  can  change  18  to  halves  and  proceed  as  fol- 
lows :  *£■  +  f  =  36  +  3. 

The  following  are  the  steps :  18  is  multiplied  by  2,  and  the  product 

18  v  2 
is  divided  by  3,  or    °  *    ,  which  is  the  same  as  18  x  f . 

o 

That  is,  18  +  |  =  18  X  f. 

3.  If  3  yards  of  dress  goods  are  required  to  make  a  waist, 
how  many  waists  can  be  made  out  of  18  yards  ? 

The  number  of  waists  =  18  -r-  3  =  \  of  18  =  18  x  \. 
That  is,  18  -?- 1  =  18  x  \. 

4.  If  an  apron  requires  \\  yards  of  material,  how  many 
aprons  can  be  made  out  of  18  yards  ? 

The  number  of  aprons  =  18  -*•  \\  =  18  -*-  \  =  18  x  f. 


Fractions.  77 

5.  If  it  takes  three-quarters  of  a  pound  of  flour  to  make 
a  loaf  of  bread,  how  many  loaves  can  be  made  with  18 
pounds  of  flour? 

The  number  of  loaves  =  18  -j-  f  =  18  x  $ . 

6.  At  three-quarters  of  a  dollar  each,  how  many  dolls  can 
be  bought  for  a  dollar  and  a  half  ? 

To  divide  by  $  (examples  1  and  3),  we  multiply  by  $. 
To  divide  by  f  (examples  2  and  4),  we  multiply  by  f. 
To  divide  by  £  (examples  5  and  6),  we  multiply  by  f. 

To  divide  by  a  fraction,  multiply  by  the  divisor  inverted. 

7.  Divide  8  by  |. 

8-=-§  =  8xf  =  10,  Ans. 

8.  Divide  f  by  10. 

9.  Divide  6f  by  9. 

142.  Divide: 

1.  |+4  4.   ^-A         7.   A-f-H  9.   t+3| 

2.  f  +  10  5.    H-t\         8.    3f  +  *  10.    i-| 

3.  lf+5  6.    A+A 

143.  "Written  Exercises. 

1.  Divide  A  by  & 

3        5 

16  *  20     J0      2       4        * 
4 

2.  Divide  15^  by  13. 

Changing  the  mixed  number  to  an  improper  fraction,  we  have, 

13 


7« 


8 

Chapter  Two. 

Divide : 

3-  A+A 

8.  A^ 

13. 

8A  +  3J 

4.    5-f-lf 

9-    tt  +  *A 

14. 

9* +  3* 

5.    8$ -5- 11 

io.  f+* 

15. 

18J  +  1H 

6.   4^-17 

ii.  A^-l 

16. 

2H+6# 

7.   24J-20 

12.  A+A 

Note.  —  The  pupil  should  prove  his  answers  to  each  of  the  foregoing 
examples  by  multiplying  the  quotient  by  the  divisor.  If  his  answer  is 
correct,  this  product  will  equal  the  dividend. 

144.  Perform  operations  indicated : 

17.  (3fx4i)-10J  24.    (tfx«)  +  (4f  x6» 

18.  (13f-7f)xf  25.   34f-17f  +  20« 

19.  (20  x  })-*-!  26.    18|  +  24^-36iJ 

20.  (20  +  f)x|  5|x9x7| 

21.  20  +  (|xf)  W'        4f  xf 

22.  (20-*-$)-s-f  51  x  73  x  31,  x  6^ 

23.  (14  j  x  7)  -  (9  x  10 1)  2fx  4^x31 

145.  Oral  Problems. 

Give  analysis  of  each  : 

1.  If  base-balls  are  worth  f  of  a  dollar  each,  what  will 
be  the  cost  of  16  base-balls  ? 

Note. — The  pupil  is  frequently  at  a  loss  to  determine  whether  a 
given  problem  in  fractions  involves  multiplication  or  division.  In 
such  a  case,  he  should  substitute  for  the  fraction  a  whole  number  to 
ascertain  the  proper  operation.  While  in  example  1  a  pupil  would 
analyze  without  hesitation :  "  If  base-balls  are  worth  $  |  each,  16 
balls  would  cost  16  times  $£,"  he  might  stumble  at  No.  2.  By  read- 
ing the  problem,  "Paid  a  certain  sum  for  base-balls  at  $3  each,"  he 
would  see  that  the  number  of  balls  is  ascertained  by  division.  His 
analysis  would  then  be,  "If  base-balls  are  $f  each,  I  can  buy  as 


Fractions.  79 

many  balls  as  there  are  $£  in  $  12."  The  work  would  be  12  -s-f  = 
12  x  f .  He  could  complete  the  solution  by  finding  $  of  12,  taking  $ 
of  12  as  4,  etc.  Another  method  of  solving  this  problem  mentally, 
is  to  change  the  price  to  a  whole  number  and  to  make  a  corresponding 
change  in  the  cost.  "Paid  4  times  $12  for  base-balls  at  4  times  $  J 
each  ;  i.e.  % 48  for  balls  at  $  3  each." 

2.  Paid  $  12  for  base-balls,  at  J  of  a  dollar  each.     How 
many  were  bought  ? 

3.  What  is  the  cost  of  2  feet  of  ribbon  at  30  cents  per 
yard? 

4.  Find  how  much  a  yard  of  ribbon  is  worth,  if  -|  yard 
costs  20  cents. 

5.  If  it  takes  f  yard  of  material  to  make  a  child's  waist, 
how  many  can  be  made  from  a  piece  containing  24  yards  ? 

6.  If  18  jackets  require  24  yards  of  cloth,  how  much  is 
needed  for  1  jacket  ? 

7.  A  man  had  60  acres  of  land.     How  many  acres  had 
he  left  after  selling  J-  of  his  land  ? 

8.  After  spending  £  of  his  money,  a  person  had  $26 
remaining.     How  much  money  had  he  at  first  ? 

9.  When  tea  is  $  .50  per  pound,  how  much  can  be 
bought  for  $  .75  ? 

10.  If  tea  is  worth  f  of  a  dollar  per  pound,  how  much 
can  be  bought  for  i  of  a  dollar  ? 

11.  When  silk  is  selling  at  $  .75  per  yard,  how  much  can 
be  bought  for  one-fourth  of  a  dollar  ? 

12.  Find  the  cost  of  a  gallon  of  milk  at  the  rate  of  9  cents 
for  3  pints. 

13.  |  of  a  gallon  of  milk  costs  9  cents.     What  is  the 
price  per  gallon  ? 

14.  f  of  what  number  is  12  ? 

15.  1  yard  and  1  foot  of  wire  cost  16  cents.     How  mucb 
must  be  paid  for  a  yard  ? 


8o  Chapter  Two. 

146.  Written  Problems. 

1.  How  much  does  a  man  earn  in  a  day  if  he  earns  45J 
dollars  in  a  month  of  26  working  days  ? 

2.  When  flour  is  5 \  dollars  per  barrel,  how  many  barrels 
can  be  bought  for  294  dollars  ? 

3.  If  coffee  is  37£  cents  per  pound,  how  many  pounds 
can  be  bought  for  60  dollars  ? 

4.  A  man  divided  16  dollars  among  some  boys,  giving 
to  each  If  dollars.     How  many  boys  received  a  share  ? 

5.  Paid  38^  dollars  for  6 J  cords  of  wood.     What  was 
the  price  per  cord  ? 

6.  How  many  steps  will  it  take  to  walk  2640  feet,  each 
step  being  2 J  feet  in  length  ? 

7.  A  man  put  40^  bushels  of  barley  into  bags  holding 
1-|  bushels.     How  many  bags  were  required  ? 

8.  In  2 J  acres  of  land,  how  many  building  lots  of  f  of 
an  acre  ? 

9.  If  |  of  a  farm  is  worth  $  8000,  what  is  {  of  it  worth? 

10.  The  product  of  two  factors  is  9^ ;  one  factor  is  3£. 
What  is  the  other  ? 

SPECIAL  DRILLS  —  REVIEW. 

147.  Give  sums  at  sight: 

1.  59  +  75  =  59  +  70  +  5  = 

2.  48  +  63  •  5.   88  +  22  8.   66  +  56 

3.  69  +  47  6.   94  +  38  9.   29  +  94 

4.  67  +  83  7.   61+39  10.   65  +  86 

11.  560  +  390  =  560  +  300  +  90  = 

12.  270  +  280  14.   430+480  16.   420  +  280 

13.  640  +  260  15.   250  +  390  17.   780  +  260 


Review.  81 

18.  225  +  154  =  225  +  150  +  4  = 

19.  315  +  421  21.   540  +  355  23.    172  +  304 

20.  437  +  260  22.   248  +  131  24.   517  +  329 

148.  Give  remainders  at  sight : 

1.  134  —  75  =  134-70-5  = 

2.  150-83  5.   124-89  8.   100-61 

3.  132-94  6.    112-56  9.    124-35 

4.  122-56  7.   180-89  10.   132-38 

11.  750  —  290  =  750-200-90  = 

12.  510-220  14.   820-560  16.   910-550 

13.  630-380  15.   730-440  17.   380-290 

18.  279  —  154  =  279-150-4  = 

19.  386-263  21.   668-325  23.   386-123 

20.  457-237  22.   279-125  24.   721-468 

149.  Give  products  at  sight: 

1 .  49  X  4  =  4  forties  +  4  nines. 

2.  47  x  3    3.  48  x  4    4.  43  x  5     5.  46  x  6    6.  38  X  7 

7.  123  X  3  =  3  times  one  twenty  three  =  three  sixty  nine. 

8.  431  x  2        10.  332  x  3        12.  232  x  3 

9.  122  x  4        11.  242  x  2        13.  31  x  24 

14.  47  X  25  =  k  of  47  hundred  =  llf  hundred  =  1175. 

15.  25  X  38  =  38  X  25  =  1  of  38  hundred  =  9f  hundred. 

16.  32  x  25  18.   44  x  25  20.   49  x  25 

17.  25  x  33  19.   25  x  45  21.   63  x  25 

150.  Give  quotients  at  sight : 

1 .  925  -f-  25 = 9J  hundred  -r-  J  hundred  =  9J  -*• 1  =  91  X  4. 

2.  875-4-25  4.   725-^-25  6.   575-4-25 

3.  625 -s- 25  5.   450-5-25  7.   350-5-25 


22  Chapter  Two. 

151.   Oral  Problems. 

1.  Find  the  cost  of  a  pound  of  tea  at  75  cents,  and  a 
piece  of  ham  at  56  cents. 

2.  A  farmer  sold  58  sheep  from  his  flock  of  121  sheep. 
How  many  remained  ? 

3.  What  will  be  paid  for  8  pounds  of  coffee  at  35^  per 
pound  ? 

4.  A  laborer  received  $  4.88  for  four  days'  work.     How 
much  did  he  earn  per  day  ? 

5.  At  $40  each,  how  many  cows  can  be  purchased  for 
$2000? 

6.  Bought  20  pounds  of  sugar  at  5^  per  pound,  and  2\ 
pounds  of  butter  at  30^.     What  was  the  amount  of  my  bill  ? 

7.  A  piece  of   cloth  measuring  31^  yards  was  divided 
into  2  equal  parts.     Find  the  length  of  each. 

8.  How  many  weeks  in  a  year  of  366  days  ? 

9.  If  I  pay  25  cents  for  3  pounds  of  cherries,  how  many 
pounds  can  I  buy  for  $1.25  ? 

10.  Find  the  cost  of  a  bushel  and  a  peck  of  peanuts  at 
the  rate  of  5  cents  per  quart. 

11.  A  farmer  had  164  acres  of  land.     How  much  had  he 
left  after  selling  87  acres  ? 

12.  Find  the  total  number  of  pounds  in  3  tubs  of  butter 
weighing  respectively  25  pounds,  34  pounds,  and  36  pounds. 

13.  At  60^  per  pound,  how  much  tea  can  be  bought  for 
$5.85? 

14.  A  drover  paid  $  219  for  oxen,  at  an  average  price  of 
$  73.     How  many  did  he  buy  ? 

15.  What  will  be  the  cost  of  20  bushels  of  wheat  at 
$1.04£  per  bushel? 

16.  At  24^  per  pound,  how  many  ounces  of  butter  can  be 
bought  for  18^? 


Review.  83 

17.  A  woman  pays  $  540  per  year  for  a  house.     What  is 
the  rent  per  month  ? 

18.  How  many  weeks  in  294  days  ? 

19.  At  72^  per  yard,  what  will  be  the  cost  of  2  ft.  11  in. 
of  lace  ? 

20.  How  much  does  a  grocer  receive  for  a  barrel  of  flour, 
196  pounds,  retailed  at  3  cents  per  pound  ? 

21.  If  47  men  can  do  a  piece  of  work  in  4  days,  how  long 
will  it  take  1  man  to  do  the  same  work  ? 

22.  Find  the  cost  of  36  acres  of  land  at  $25  per  acre. 

23.  If  it  takes  3^  yards  of  cloth  to  make  a  coat,  how 
many  coats  can  be  made  from  24J  yards  ? 

24.  How  much  will  be  paid  for  84  yards  of  silk  at  $lf 
per  yard  ? 

25.  If  a  certain  quantity  of  provisions  will  last  one  man 
215  days,  how  long  will  it  last  43  men  ? 

26.  How  many  square  yards  are  there  in  a  rectangular 
field  36  yards  long  and  25  yards  wide  ? 

152.   Written  Exercises. 

1.  What  is  the  sum  of  94,625;  215;  5172;  819,365; 
121? 

2.  Bought  172  acres  of  land  for  $860.  What  was  that 
an  acre  ? 

3.  In  a  classroom  there  are  54  pupils ;  each  pupil  spent 
$  2.75  for  books  this  year.  How  much  money  was  spent 
for  books  by  the  whole  class  ? 

4.  By  the  census  of  1890,  Massachusetts  had  a  popula- 
tion of  2,238,943 ;  in  1900,  it  had  a  population  of  2,805,346, 
What  was  the  gain  ? 

5.  How  many  boxes  of  strawberries  at  $.15  a  box  can 
I  get  for  $  1.20  ? 


84  Chapter  Two. 

6.  What  is  a  proper  fraction?     An  improper  fraction? 
Define  numerator,  denominator,  a  mixed  number. 

7.  Add  J,  |,  |,  and  ± 

8.  If  7  pairs  of  shoes  cost   $12J,  how  much  will  one 
pair  cost  ? 

10.  What  is  the  product  of  -fa,  ^,  if,  and  f$  ? 

11.  8ix7|  =  ? 

12.  Paid  |  of  a  dollar  for  potatoes,  %  of  a  dollar  for 
apples,  and  ^  of  a  dollar  for  sugar.  How  much  did  I  pay 
for  all  ? 

13.  Divide  2\  by  If 

14.  Find  the  difference  between  4§  and  3f. 

MULTIPLICATION  OF  DECIMALS. 
153.   Oral  Problems. 

1.  When  the  French  franc  is  worth  19.3  cents,  what  is 
the  value  of  the  20-f ranc  piece  in  United  States  money  ? 

2.  What  is  the  equivalent  of  10  German  marks,  the  mark 
being  quoted  at  23^-  cents  ? 

3.  A  man  paid  100  pounds  sterling  for  a  piano.     Find 
the  cost  in  U.  S.  money  at  $  4.8665  per  pound  sterling. 

Note.  —  $  4.8665  may  be  read  4  dollars  86  cents  6  mills  and  5  tenths 
of  a  mill,  a  mill  being  one-tenth  of  a  cent. 

4.  A  meter  contains  39.37  inches.     How  many  inches  in 
100  meters  ? 

5.  One  kilogram  =  2.2046  pounds.    What  is  the  equiva- 
lent of  1000  kilograms,  in  pounds  ? 

Note.  —.2046  is  read  2046  ten-thousandths. 

6.  How  many  square  yards  are  there  in  a  piece  of  ground 
40  yards  long  and  12.5  yards  wide  ? 


Decimals.  85 

* 

7.  How  many  ounces  in  2.5  pounds  ? 

8.  Change  .75  hour  to  minutes. 

9.  Find  the  perimeter  of  a  square,  each  side  of  which 
measures  10.25  feet. 

154.  Written  Exercises. 

1.  Multiply  38.4  by  6.37. 

Place  the  units'  figure  (6)  of  the  multiplier  under  38.4 
the  last  figure  (4)  of  the  multiplicand.     6  times  6.37 

4  tenths  =  24  tenths  =  2.4  ;  write  .4  under,  the  multi-  230.4 

plier  6,  and  carry  2  ;    etc.    Next  multiply  by  .3,  11.52 
or  fV    A  x  t%  =  T<fr>  or  -12-    Write  2  in  the  hun-  2.688 

dredths'  place,  and  carry  1  tenth;   etc.    Multiply  244  f 08 
finally  by  .07,  or  T^.     &  x  T%  =  T$fo,  or  .028. 
Write  8  in  the  thousandths'  place,  etc.                       Ans-    *44-w°- 

Note.  —  By  writing  the  units'  figure  of  the  multiplier  under  the  last 
figure  of  the  multiplicand,  and  by  taking  care  to  place  the  right-hand 
figure  of  each  partial  product  under  the  corresponding  figure  of  the 
multiplier,  the  decimal  points  in  the  partial  products  and  the  total  will 
naturally  fall  under  the  decimal  point  in  the  multiplicand. 

2.  Multiply  12.34  by  56.7. 

12.34  While  pupils  should  occasionally  begin  to  ^2  34 

gg  j  multiply  by  the  left-hand  figure  (5)  of  the  gg  j 

n-ij  r\  multiplier,  some  may  prefer  to  begin  with  the  ft^RQft" 

74  04.  right-hand  figure  (7).     It  will  be  noted  that  „.  \. 

8  £oo  the  number  of  decimal  places  in  the  product  n^r, \ 

,,„„''  equals  the  sum  of  those  in  the  multiplier  and      „     *„  r 

699.678      2       u.  ..      .  *  699.678 

the  multiplicand. 

Multiply  as  in  ivhole  numbers,  and  from  the  right  of  the 
product  point  off  as  many  decimal  places  as  there  are  decimal 
places  in  both  factors. 

155.  Multiply: 

1.  32x2.5  3.    6.4x4.5 

2.  3.2x25  4.   7.2x3.75 


16. 

18.4  x  20.25 

17. 

11.16  X  42.40 

18. 

66.6  x  3.3£ 

19. 

6.24  x  1.75 

20. 

400.04  x  39.25 

86  Chapter  Two. 

5.  12.8  x  5.7  8.   5.625  x  8.4 

6.  9.6  x  1.125  9.   1.875  x  12.8 

7.  34.9x2.34  10.   42.36x2.95 

Note.  —  The  pupil  should  correct  any  error  he  may  make  in  placing 
the  decimal  point  by  estimating  the  approximate  answer.  The  answer 
to  example  3,  for  instance,  is  more  than  2  times  32  and  less  than  3  times 
32.    In  example  3,  it  is  more  than  4  sixes  and  less  than  6  sevens. 

11.  1.75  x  64 

12.  8.375  x  40 

13.  24.5  x  18.2 

14.  9.6  x  12£ 

15.  7.43  x  3.6 

156.   Oral  Problems. 

1.  I  owned  40  acres  of  land  and  sold  .25  of  it.     How 
many  acres  did  I  sell  ? 

2.  A  boy  bought  15  hens,  which  was  .6  of  what  he  al- 
ready had.     How  many  had  he  at  first  ? 

3.  A  lawyer  charged  me  .11  of  the  money  for  collecting 
$  100.     How  many  dollars  did  he  charge  ? 

4.  If  I  earn  $  8  in  a  week,  how  much  can  I  earn  in  7.5 
weeks  ? 

5.  .75  of  a  class  of  44  were  promoted.     How  many  were 
not  promoted  ? 

6.  What  is  the  surface  of  a  table  4  feet  wide  and  6.25 
feet  long  ? 

7.  .5  of  a  yard  is  how  many  feet  ?     How  many  inches  ? 

8.  A  man  bought  3.5  yards  of  cloth  at  $  5  a  yard.     What 
was  the  price  ? 

9.  25  miles  is  .5  of  the  distance  between  two  cities. 
What  is  the  distance? 

10.  In  a  box  were  100  oranges;  .08  of  them  became 
spoiled.     How  many  sound  ones  were  left  ? 


Decimals.  87 

157.  Written  Problems. 

1.  How  many  yards  are  there  in  25  pieces  of  carpeting 
if  each  piece  contains  32.75  yards  ? 

2.  A  mill  uses  95.6  tons  of  coal  per  day.  How  many 
tons  will  it  use  in  42.25  days  ? 

3.  A  cubic  foot  of  water  weighs  62.5  pounds ;  ice  is  .92 
as  heavy  as  water.  What  is  the  weight  of  a  cubic  foot 
of  ice  ? 

4.  I  bought  3  loads  of  wood,  the  first  containing  1.04 
cords,  the  second  1.05  cords,  and  the  third  .946  cord. 
What  did  it  cost  me  at  $  4.50  a  cord  ? 

5.  A  gallon  of  water  weighs  8.33  pounds.  What  is  the 
weight  of  a  gallon  of  milk  which  is  1.03  times  as  heavy  as 
water  ? 

6.  A  wheel  in  making  one  revolution  travels  15.03  feet. 
How  far  will  it  travel  in  25  revolutions  ? 

7.  A  ship  sails  18.54  miles  in  an  hour.  How  far  will  she 
sail  in  15.5  hours  ? 

^.  Find  the  cost  of  concreting  a  cellar  24.5  feet  long  by 
14.25  feet  wide,  at  30  cents  per  square  foot. 

9.  A  quantity  of  provisions  will  last  25  men  12.75  days. 
How  long  will  it  last  one  man  ? 

10.  Two  men  start  from  the  same  place  and  travel  in  op- 
posite directions,  one  at  the  rate  of  3.85  miles  per  hour,  and 
the  other  at  the  rate  of  4.12  miles  per  hour.  How  far  apart 
will  they  be  at  the  end  of  13  hours  ?     Make  a  diagram. 

DIVISION  OF  DECIMALS. 

158.  Divide  42  by  2.1. 

Changing  the  decimal  fraction  in  the  divisor  to  a  common  fraction, 
we  have  42  -2^  =  ^-H=  ^  x  tf  =  *&> 

42-2.1  =  420-7-21. 

Note.  — When  we  change  the  divisor  2.1  to  21,  we  have  multiplied 
it  by  10,  and  the  same  change  must  be  made  in  the  dividend. 


88  Chapter  Two. 

Make  the  divisor  a  wJiole  number,  and  make  a  correspond- 
ing change  in  the  number  of  decimal  places  in  the  dividend. 
This  reduces  the  numbers  to  the  same  denomination.  If  neces- 
sary to  complete  the  operation,  ciphers  may  be  annexed  to  the 
dividend.  The  number  of  decimal  places  in  the  quotient  is 
equal  to  the  number  in  the  dividend  as  changed. 


9.  50  -r-  .25 

10.  72  -  .5 

11.  960  -  .03 

12.  '.847 -.007 

13.  27  -=-  .002 

14.  10  -r-  .8 

15.  1.263 -f- .03 

16.  19.63 -5- .013 

17.   Diyide  196.3  by  .013. 

Remove  the  decimal  point  in  the  divisor  three 
places  to  the  right,  and  make  a  corresponding 
change  in  the  dividend,  adding  two  ciphers. 

To  show  where  the  decimal  point  originally  be- 
longed, draw  a  cancellation  mark  through  it, 
instead  of  erasing  it. 

When  the  divisor  is  thus  made  a  whole  number,  the  decimal  point 
In  the  quotient  will  be  placed  under  (or  over)  the  new  decimal  point 
In  the  dividend. 

1.736  -  16  17.36  -j-  .16  .01736  -i- 1.6 

.1085  Ans.  108.5  Ans.  .01085  Ans. 


159.   Written  Exercises. 

Divide : 

1. 

80-2.5 

2. 

8+2.5 

3. 

840  -f- 1.2 

4. 

se^A 

5. 

36 -.9 

6. 

12.6  -f-  6.3 

7. 

48  +  15 

8. 

18.36  +  .6 

L6)1.7360  /16.)17/36.0  l/6.)/0.17360 


Decimals. 

18. 

.504 --.024 

26. 

392  --  3.2 

19. 

5.04  --  .24 

27. 

48  -5- 3000 

20. 

50.4  --  2.4 

28. 

92  -*-  .23 

21. 

504  -5-  24 

29. 

.875  - 125 

22. 

168  -s-,7 

30. 

381.17  -f-  8.11 

23. 

36  -5-  .12 

31. 

.624  -r-  9.75 

24. 

.875  -5-  .25 

32. 

48.195  -5-  3.57 

25. 

123.6  -v-  .01 

33. 

829.31  -5-  .019 

89 


160.   Divide  381.6  by  95.032. 


Note. — The  sign  (+)  after  the  last 
figure  of  the  quotient  indicates  that  there 
is  a  remainder. 


4.015  + 


95/032.)381/ 600.000 
380128 
147200 
95032 


521680 


161.   Divide,  carrying  out  the  quotient  to  3  places  of 


decimals 

34.  31  -5-  13 

35.  4.5  -5- 17 

36.  920.07-5-46 

162.   Write  answers  at  sight : 


37.  7.049-5-1.6 

38.  81.22-5-3.275 

39.  246.3  -5-  93.473 


Note. — To  multiply  .042  by  100,  the  decimal  point  is  moved  two 
places  to  the  right ;  i.e.  .042  x  100  =  4.2  ;  .042  x  200  =  4.2x2  =  8.4. 


1.  .042  x  200 

2.  .13  x  300 

3.  .014x50 

4.  8.1  x  60 


5.  40  x  .7 

6.  25  x  .8 

7.  234  x. 2 

8.  .73  x  30 


9.  .121x4000 

10.  .061x500 

11.  .03x1000 

12.  .012  x  700 


Note.  —  Remember  that  369  -=- 1000  =  T8^  =  369  thousandths =.369, 
To  divide  369  by  3000,  therefore,  divide  .369  by  3. 


90  Chapter  Two. 

13.  369-^-3000  17.  2460  +  3000-  21.  4.68  +  20 

14.  219  +  300  18.  196  +  4000  22.  30.5  +  500 

15.  48.6  +  60  19.    6  +  500  23.  18.8  +  200 

16.  1.89  +  90  20.  27.9  +  300  24.  248  +  4000 

163.  Written  Exercises. 

1.   1728  +  1200        2.    172.8  +  1200        3.   1.728  +  1200 

1200)17.28/  1200)1.72/8  1200).Ol/728 

1.44  Ans.  .144  Ans.  .00144  Ans. 

Cancel  the  ciphers  in  the  divisor,  and  remove  the  decimal 
point  in  the  dividend  a  corresponding  number  of  places  to  the 
left,  prefixing  ciphers  if  necessary. 

164.  Divide: 

1.  2436  +  3000  7.   45  +  800 

2.  136.5  +  1300  .  8.  25.2  +  240 

3.  84.8  +  80  9.  345.6  +  1200 

4.  100.1  +  700  10.  4004  +  110 

5.  1  +  40  11.  5.28  +  60 

6.  2.2  +  50  12.  907.5  +  1500 

165.  Oral  Problems. 

1.  I  cut  8.72  yards  of  cloth  into  8  equal  pieces.     How 
long  was  each  piece  ? 

2.  I  divided  .75  of  a  pound  of  candy  equally  among  3 
girls.     What  part  of  a  pound  did  each  receive  ? 

3.  I  divided  .5  of  a  pound  of  cherries  among  4  children. 
What  part  of  a  pound  did  each  receive  ? 

4.  49  rods  is  .7  of  the  distance  round  a  field.     How 
many  rods  of  fence  will  enclose  the  field? 

5.  24  yards   of  matting  cover  .8  of  my  floor.     How 
many  yards  more  must  I  buy? 


Decimals.  91 

6.  40  pounds  are  .4  of  my  weight.     What  do  I  weigh  ? 

7.  I  spent  2.5  dollars,  which  was  .5  of  what  I  had.    How 
much  had  I  ? 

8.  36  square  inches  are  .25  of  a  square  foot.     How  many- 
square  inches  in  a  square  foot  ? 

9.  A  collector  receives  .05  of  all  the  money  he  collects. 
How  much  did  he  collect  to  earn  9 15  ? 

10.   At  75  cents  each,  how  many  chairs  can  be  bought 
for  $12? 

166.  Written  Problems. 

1.  If  35.84  cubic  feet  of  water  weigh  a  ton,  what  will  be 
the  weight  of  2458.6  cubic  feet  ?  * 

2.  How  many  francs  are  there  in  $150?      (A  franc 
equals  19.3^.) 

3.  If  a  barrel  of  flour  costs  $5.75,  how  many  barrels  can 
be  bought  for  $  258.75  ? 

4.  If  $  640.05  are  paid  for  75.3  tons  of  coal,  what  is  the 
price  per  ton  ? 

5.  There  are  31.5  gallons  in  a  barrel.     How  many  bar- 
rels are  there  in  2787.75  gallons  ? 

6.  I  have  96  cubic  feet  of  wood ;  this  is  .75  of  a  cord. 
How  many  cubic  feet  in  1  cord  ? 

7.  A  man  earns  $162  in  13.5  weeks.     What  are  his 
wages  per  week? 

8.  I  bought  a  farm  of  71.5  acres  for  $6220.50.     What 
did  it  cost  me  per  acre  ? 

9.  There  are  2150.4  cubic  inches  in  a  bushel.      How 
many  bushels  are  there  in  9676.8  cubic  inches  ? 

10.   The  wheel  of  a  bicycle  is  7.25  feet  around.      How 
many  times  will  it  turn  in  going  a  mile,  or  5280  feet  ? 


gi  Chapter  Two. 

UNITED  STATES  MONEY. 
FRACTIONAL  PARTS  OF  A  DOLLAR. 

167.   Oral  Problems. 

1.  How  many  50-cent  base-balls  can  be  bought  for  f  15  ? 

(15  -r- 1,  i.e.  15  x  2) 

2.  How  many  75-cent  base-balls  can  be  bought  for  $  15  ? 

(15  h-  f,  i.e.  15  x  |) 

3.  At  75^  per  pound,  how  much  tea  can   be  bought 
for  $  1  ? 

4.  How   many  hats,   at   $1.25   each,   can   be   bought 
for  $15?  (15  ^-1|) 

5.  Paid  $16  for  coffee  at  25^  per  pound.     How  many 
pounds  were  purchased  ? 

6.  At  33^  per  pound,  how  many  pounds  of  butter  can 
be  bought  for  $  32  ? 

7.  Find  the  number  of  yards  of  ribbon,  at  12-J-^  per  yard, 
that  will  cost  $45. 

8.  At  6J£  per  bar,  how  many  bars  of  soap  will  cost  $  11  ? 

9.  If  4  pieces  of  violet  soap  are  sold  for  25^,  how  many 
can  be  bought  for  $  9  ? 

10.  $  24  is  paid  for  corn  at  75^  per  bushel.     How  many 
bushels  are  bought  ? 

11.  I  spent  $30  for  lace  at  66^  per  yard.     How  many 
yards  did  I  buy  ? 

12.  For  $  36  how  many  pairs  of  rubber  shoes  can  be 
bought  at  37^  per  pair? 

13.  Oats  are  62^  per  bushel.     How  many  bushels  will 
$40  buy? 

14.  A  farmer  pays  87^  per  bushel  for  seed  rye.     If  his 
bill  amounted  to  $  21,  how  many  bushels  did  he  purchase  ? 


United  States  Money.  93 

15.  A  storekeeper  sold  $  33  worth  of  collars,  at  16f  ^ 
each.     How  many  did  he  sell  ? 

16.  At  the  rate  of  3  for  50^,  how  many  collars  can  be 
bought  for  $25? 

17.  Corn  is  worth  20^  per  can.      How  many  cans  will 
cost  $  32  ? 

18.  Find  the  cost  of  35  yards  of  cloth,  at  $  1.25  per  yard. 

19.  At  $  1.25  per  yard,  how  many  yards  of  cloth  can  be 
bought  for  $  35  ? 

20.  How  many  pairs  of  gloves,  at  $  1.75  per  pair,  will 
cost  $  28  ? 

21.  When  coal  is  $  5.25  per  ton,  how  many  tons  can  be 
bought  for  $  42  ? 

22.  Cost  of  16  pairs  of  shoes  at  $  2.75  ? 

23.  33  jackets  at  $  3.33J  ?    24.   18  yards  cloth  at  $  2.16|  ? 

25.  Paid  $  26  for  cloth  at  $  2.16|  per  yard.     How  many 
yards  did  I  buy  ? 

26.  Find  the  cost  of  16  pairs  of  skates  at  $  1.87^-  per  pair. 

27.  If  sheep  cost   $3. 12^  each,  how  many  can  I  get 
for  $  75  ? 

28.  How  many  25-cent  balls  can  be  bought  for  $  8.75  ? 

29.  Divide  775  by  25.  30.   Divide  $8.25  by  75^. 

31.  How  many  square  feet  are  there  in  a  lot  96  feet  long, 
100  feet  wide  ?     In  a  lot  96  feet  long,  25  feet  wide  ? 

32.  Find  the  total  cost  of  32  head  of  cattle  at  $  75  per 
head. 

33.  How  much  must  be  paid  for  32  cows  at  $37.50  each  ? 

34.  If  sheep  are  worth  $3.75  each,  how  much  will  a 
farmer  receive  for  32  sheep  ? 

35.  If  a  train  goes  at  the  rate  of  25  miles  per  hour,  how 
many  hours  will  it  take  to  go  675  miles  ? 


16  oz. 
Xl7 

112 
16 

272  oz. 
Add      4  oz. 

276  oz. 
4qt. 

37  gal.  3  qt. 

151  qt. 

94  ,       Chapter  Two. 

DENOMINATE  NUMBERS. 

Note.  — For  the  tables  of  Denominate  numbers  used  in  these  les- 
sons, see  section  93,  pages  43-44. 

168.   Written  Exercises. 

1.   Change  17  lb.  4  oz.  to  ounces. 


Since  there  are  16  ounces  in  1  pound,  in 
17  pounds  there  are  272  ounces,  etc. 

Add      4  oz. 

Arts. 

2.  Change  37  gal.  3  qt.  to  quarts. 
In  this  example,  we  are  to  multiply  4 

quarts  (the  number  in  a  gallon),  by  37,  and 

to  add  3  quarts  to  the  product.     In  practice,  \§±  q^     Ans. 

however,  4  is  taken  as  the  multiplier,  and 

the  three  quarts  are  added  in.     We  say  4  sevens  are  28,  and  3  are  31, 

writing  the  1 ;  4  threes  are  12,  and  3  are  15. 

3.  Change  45  bushels  to  quarts. 

Write    as    here    shown,    placing  4  pk.  8  qt. 

above  0  pecks  the  number  of  pecks     45  ku.  Opk.  0  qt. 

in  a  bushel,  and  above  0  quarts  the  jca  pk#     1440  qt. 

number  of  quarts  in  a  peck.     Multi-  .       ..  1Ati     , 

ply  4  pecks  by  45,  and  write  the 
product,  180  pecks,  in  the  proper  column ;  multiply  8  quarts  by  180,  etc. 

Change : 

4.  63  qt.  1  pt.  to  pints. 

5.  27  bu.  3  pk.  to  pecks. 

6.  48  pk.  7  qt.  to  quarts. 

7.  84  pk.  to  pints. 

8.  7  mi.  60  rd.  to  rods. 

9.  13  hr.  20  rnin.  to  minutes 
&0.  18  wk.  3  da.  to  days,, 


Denominate  Numbers.  95 

11.  Change  151  quarts  to  gallons  and  quarts. 
Write  above  151  quarts  the  number  of  quarts  4  Q^ 

in  a  gallon.    Divide  151  by  4  to  obtain  the  num.-  15T~ot~ 

ber  of  gallons,  37.     Write  the  remainder,  3,  in    ^Z ;~ 7; — 7~    A 

f.       . 6        .'  '    '        37  gal.  3  qt.  An& 

the  column  of  quarts. 

12.  Change  228  inches  to  yards  and  feet. 

Divide  the  number  of  inches,  228,  by  12,  to  3  f^  ^2  in 

obtain  the  number  of  feet,  19.     Write  this  to  -in  ft  228  in 

the  left  of  228  inches.    Reduce  to  yards  by  divid-  a  va    1  f 4-  An* 
ingby3. 

13.  87  pints  to  quarts  and  pints. 

14.  250  feet  to  yards  and  feet. 

15.  1650  rods  to  miles  and  rods. 

16.  864  hours  to  weeks. 

17.  296  quarts  to  bushels  and  pecks. 

18.  315  ounces  to  pounds  and  ounces. 

19.  743  months  to  years  and  months. 

20.  15,000  minutes  to  days  and  hours. 

21.  Add  3  ft.  6  in.,  9  ft.  5  in.,  12  ft.  3  in. 

Write  the  feet  in  one  column  and  the  inches      3  ft.  6  in. 
in  another.     The  sum  of  the  column  of  inches  is      9  ft.  5  in. 

14  inches,  or  1  foot  2  inches.     Write  2  inches,  12  ft.  3  in. 

and  carry  1  foot  to  the  next  column.  25  ft.  2  in.  Ans. 

22.  30  min.  15  sec.  +  30  min.  18  sec.  +  45  min.  24  sec. 

23.  9  yr.  3  mo.  -f  18  yr.  7  mo.  +  22  yr.  2  mo. 

24.  19  wk.  4  da.  +  7  wk.  5  da.  +  8  wk. 

25.  9  mi.  169  rd.  +  84  rd.  -f-  3  mi.  67  rd. 

26.  7  yd.  1  ft.  +  33  yd.  -f- 19  yd.  2  ft 

27.  18  gal.  1  qt.  + 16  gal.  2  qt.  +  15  gal.  3  qt 

28.  5  pk.  3  qt.  +  6  qt.  +  7  pk.  1  qt. 


$6  Chapter  Two. 

29.  24  bu.  3  pk.  +  24  bu.  3  pk.  +  24  bu.  3  pk. 

30.  12  qt.  1  pt.  + 12  qt.  1  pt.  + 12  qt.  1  pt.  + 12  qt.  1  pt 

31.  Multiply  12  qt.  1  pt.  by  7. 

7  times  1  pint  =  7  pints =3  quarts  1  pint.    Write  12  at  1  pt 
1  pint  in  the  proper  column,  and  carry  3  quarts.  „  n 

7  times  12  quarts  =  84  quarts.     Carrying  3,  we  <*»     7   -j      ,       a 
get  87  quarts. 

32.  12  qt.  1  pt.  x  4.  37.  15  wk.  3  da.  x  5. 

33.  24  bu.  3  pk.  x  3.  38.  7  yr.  3  ino.  x  10, 

34.  5  pk.  3  qt.  x  9.  39.  40  min.  35  sec.  x  2. 

35.  18  gal.  1  qt.  x  8.  40.  9  ft.  5  in.  x  12. 

36.  33  yd.  1  ft.  x  6. 

41.  From  25  ft.  3  in.  take  18  ft.  7  in. 

Take  7  inches  from  1  foot  3  inches,  or  25  ft.  3  in. 
15  inches.  Carry  1  foot  to  18  feet,  making  —  18  ft.  7  in. 
19  feet,  etc.  6  ft.  8  in.  Ans. 

42.  50  min.  13  sec.  —  27  min.  30  sec. 

43.  12  yr.  1  mo.  —  5  yr.  11  mo. 

44.  50  wk.  4  da.  -  18  wk.  6  da. 

45.  25  ft. -18  ft.  7  in. 

46.  33  yd.  1  ft.  -  18  yd.  2  ft. 

47.  240  gal.  1  qt.  -  94  gal.  2  qt. 

48.  83  pk.  3  qt.  -  59  pk.  1  qt. 

49.  170  bu.  1  pk.  -  85  bu.  2  pk. 

50.  135  qt.  1  pt.  -  67  qt.  1  pt. 

51.  Divide  87  gal.  2  qt.  by  5. 

Dividing  87  gallons  by  5,  we  get  17  gallons, 
and  2  gallons  remainder.     Change  2  gallons   5)87  gal.  2  qt. 
to  8  quarts,  add  in  2  quarts,  making  10  quarts.        17  gal.  2  qt.   Ans. 
Dividing  10  quarts  by  5,  we  get  2  quarts. 


Denominate  Numbers.  97 

52.  50  min.  35  sec.  -7-  5  57.  387  gal.  -*-  6 

53.  156  yr.  9  mo.  ^  9  58.  222  bu.  3  pk.  -*-  9 

54.  73  wk.  2  da.  ^  3  59.  150  qt.  -*-  4 

55.  50  mi.  135  rd.  -f-  7  60.  75  bu.  -f-8 

56.  253  yd.  1  ft.  --  10 

61.  Divide  87  qt.  by  43  qt.  1  pt. 

Change    the   divisor   to   87    pints.     43  ^  ^  p^  \  gj  ^ 
Change  the  dividend  to  the  same  de-  g<r     ^  )\J4.  Dt 

nomination.    87  pints  is  contained  2  a  Ans. 

times  in  174  pints. 

62.  50  min.  35  sec.  -r-  10  min.  7  sec. 

63.  78  bu. -j- 9  bu.  3  pk. 

64.  5  lb.  1  oz.  -f-  9  oz. 

65.  14  ft.  2  in.  -s- 1  ft.  5  in. 

169.   Oral  Exercises. 

1.  3  pints  is  what  part  of  a  gallon? 

(3  pints  is  what  part  of  8  pints  ?  ) 

2.  What  part  of  a  gallon  is  1  qt.  1  pt.  ? 

3.  Find  the  ratio  of  £  to  |. 

(Divide  10  fifteenths  by  9  fifteenths.) 

4.  Find  the  ratio  of  f  to  f . 

5.  How  many  square  feet  in  a  rectangle  12  feet  long,  13 
feet  wide? 

6.  J  of  a  day  is  how  many  hours  and  minutes  ? 

7.  14  ounces  is  what  part  of  2  pounds  ? 

8.  I  foot  is  what  part  of  a  yard  ? 

9.  A  strip  of  tape  3  yards  long  is  cut  into  four  equal 
pieces.     How  many  feet  and  inches  are  there  in  each  piece? 

10.   At  $  30  per  month,  how  much  rent  will  be  paid  in 
i  year,  8  months  ? 


98  Chapter  Two. 

11.  2 \  months  is  what  part  of  a  year  ? 

12.  At  -f  of  a  dollar  per  pound,  how  much  tea  can  I  get 
for  $1? 

13.  How  many  square  yards  in  a  room  15  feet  long,  18 
feet  wide  ? 

14.  A  lot  is  25  feet  by  100  feet.  How  many  feet  of 
fence  will  it  take  to  enclose  it  ? 

15.  1  pk.  1  qt.  is  what  part  of  a  bushel  ? 

16.  15  is  what  part  of  4  dozen  ? 

17.  Reduce  ff  to  lowest  terms. 

170.  Written  Problems. 

1.  Add  4  da.  6  hr.,  9  da.  11  hr.,  3  da.  7  hr. 

2.  What  part  of  a  week  is  1  da.  18  hr.  ? 

3.  If  a  man  receives  $  60  interest  per  year,  how  much 
will  he  receive  in  3  yr.  1\  mo.  ? 

4.  Reduce  3  da.  18  hr.  to  minutes. 

5.  How    many   days    and    hours    are    there   in    8100 
minutes  ? 

6.  -ff^  of  a  day  is  how  many  hours  ? 

7.  How  many  hours  and  minutes  in  .4  day  ? 

8.  A  man  receives  $1456  per  year  of  52  weeks.     What 
is  his  salary  per  week? 

9.  Find  the  cost  of  1  bu.  1  pk.  1  qt.  of  potatoes  at  8 
cents  per  half-peck. 

10.  A  piece  of  meat  weighing  27  lb.  12  oz.  is  divided 
among  6  persons.  How  many  pounds  and  ounces  does 
each  receive? 

11.  How  many  bushels  and  pecks  are  there  in  5  bags, 
each  containing  1  bu.  1  pk  ? 

12.  How  many  gallons,  quarts,  and  pints  of  ice-cream 
will  be  needed  to  give  a  half-pint  to  each  one  of  67  persons? 


Measurements.  99 

13.  Find  the  cost  of  7  lb.  10  oz.  of  tea  at  40  cents  per 
pound. 

14.  From  a  bin  containing  20  bushels  of  wheat  there 
were  sold  10  bu.  3  pk.     How  much  remained  ? 

15.  How  many  yards  in  a  mile  ?  How  many  feet  ?  How 
many  inches  ? 

16.  A  field  is  16  rods  long,  12  rods  wide.  How  many 
square  yards  does  it  contain?  What  is  the  perimeter  in 
rods?     In  feet? 

17.  How  many  rails  each  30  feet  long  will  be  needed  for 
a  single  track  road  (two  tracks)  40  miles  long  ? 

18.  A  boy  steps  33  inches.  How  many  steps  will  he  take 
in  going  2  miles  ? 

19.  December  20  the  sun  rises  at  Boston  at  7.26  a.m.  and 
sets  at  4.30  p.m.  How  long  is  it  between  sunrise  and  sun- 
set? How  much  longer  is  the  day  at  Charleston,  S.  C, 
where  the  sun  rises  at  6.58  a.m.  and  sets  at  4.57  p.m.  ? 

20.  On  June  21  the  sun  rises  at  Boston  at  4.23  a.m.  and 
sets  at  7.40  p.m.  On  the  same  day  it  rises  at  Charleston  at 
4.53  a.m.  and  sets  at  7.11  p.m.  What  is  the  length  of  the 
day  at  each  place  ? 

MEASUREMENTS. 

171.  How  many  square  yards  in  a  floor  6  yards  long,  5 
yards  wide  ? 

How  many  square  yards  in  a  ceiling  18  feet  long,  15  feet 
wide? 

172.  Written  Exercises. 

1.  How  many  square  yards  are  there  in  a  piece  of 
ground  60  feet  long  and  30  yards  wide  ? 

60  feet  =  20  yards.  In  a  plot  20  yards  by  30  yards  the  area  =  1 
square  yard  x  20  x  30  =  600  square  yards,  Ans. 


ioo  Chapter  Two. 

Calculate  the  number  of  square  yards  in  the  following. 
First  reduce  each  side  to  yards. 

2.  18  yd.  by  21  yd.  7.    33  ft.  by  36  yd. 

3.  54  ft.  by  63  ft.  8.   27  ft.  by  96  ft. 

4.  72  in.  by  108  in.  9.    54  ft.  by  72  in. 

5.  19  yd.  by  47  yd.  10.   48  ft.  by  45  ft. 

6.  67  yd.  by  89  yd.  11.    54  in.  by  72  ft. 
First  indicate  the  operations  ;  then  cancel. 

12.  Find  the  number  of  square  yards  in  a  room  18  ft 
4  in.  long,  22  ft.  6  in.  wide. 

18  ft.  4  in.  =  18 1  ft.  =  Mi  yd.  =  ^  yd. 
3  3  9 

22  ft.  6  in.  =  22}  ft.  =  ^  yd.  =  ^  yd. 

O  D 

5 

Area  =  1  sq.  yd.  x  5§  x  ** .     Canceling,   §L*i2  =  ?75  =  45| 

96  JfxG         6  * 

Ans.  45 1  sq.  yd. 

13.  How  many  square  yards  in  a  room  13  ft.  1  in.  long, 
27  ft.  wide  ? 

13  ft.  1  in.  =  157  in.  =  ^  yd.     27  ft.  =  9  yd. 

.Area=1s,yd.xfx0.80      ^f-«* 

4  Ans.  39£  sq.  yd. 

14.  How  many  square  inches  in  12  panes  of  glass,  each  5 
inches  long,  7  inches  wide  ? 

15.  A  piece  of  cloth  is  48  yards  long,  24  inches  wide. 
How  many  square  yards  does  it  contain  ? 

16.  A  merchant  imports  8  pieces  of  cloth,  36  yards  to  the 
piece.  How  many  square  yards  of  cloth  are  there,  if  it  is 
32  inches  wide  ? 

17.  A  tight  board  fence  6  feet  high  surrounds  a  lot  25  feet 
front  by  100  feet  deep.  How  many  square  feet  of  boards 
in  the  front  fence  ?  In  the  back  fence  ?  In  each  side 
fence  ?     In  the  whole  ?     (Make  diagrams.) 


Bills. 


101 


18.  A  room  is  18  feet  long,  15  feet  wide,  l/>  feet-  high. 
How  many  square  feet  in  the  floor  ?  *.'.•?•>•?   »* ,  ,• 

Draw  a  rectangle  to  represent  the  ceiling.  Write  the  dimensions  in 
their  proper  places,  and  write  in  the  centre  the  number  of  square  feet 
in  its  surface.  Draw  diagrams  of  the  four  walls ;  give  dimensions 
and  surface  of  each. 

19.  How  many  faces  has  a  cube  ?  If  one  edge  of  a  cube 
measures  4  inches,  how  many  square  inches  are  there  in 
the  entire  surface  ? 

Suppose  you  wish  to  make  a  cube  out  of  a  single  piece  of  paste- 
board. Make  a  drawing  to  show  the  shape  of  the  piece  needed, 
without  allowing  anything  for  overlapping  parts. 

20.  The  United  States  government  charges  a  duty  of  4^ 
per  square  yard  on  imported  cotton  cloth.  What  duty  must 
the  importer  pay  on  a  piece  containing  24  yards,  f  yard 
wide? 

21.  What  will  be  the  cost  at  $  1  per  square  yard  for 
making  a  sidewalk  12  feet  wide  and  30  feet  long? 

BILLS. 
173.  New  York,  Oct.  I,  1904. 

Mrs.  William  Johnson, 

Bought  of  Furey  &  Company. 


1904 

Aug. 

18 
15 

19 

1,4  yd.  Carpet 
8  Oak  Chairs 
1  Rocker 

18  yd.  Oil-cloth 

f  .90 

1.75 

.50 

12 

— 

27 

1  Parlor  Suit 

75 

— 

Sept. 

19 

6  Kitchen  Chairs 

.75 

1  Table 

4 

50 

26 

86  yd.  Matting 

M 

9 

io2  Chapter  Two. 

,1.-  Copy 'jthe  .  bill  on  the  preceding  page.     Supply  the 
missing  amounts. 

2.  Charles  W.  Wise  has  bought  the  following  goods  of 
Thos.  F.  Farley  &  Co. : 

Jan.  3,  1904,  50  pounds  of  sugar,  at  h\$ ;  4  pouDds  of  tea, 
at  62-£/.  Jan.  4,  10  pounds  of  coffee,  at  32|^ ;  2  barrels  of 
flour,  at  %  5.75.  Jan.  9,  24  bars  of  soap,  at  16|^  ;  42  pounds 
of  starch,  at  8^. 

Make  out  a  bill  dated  Feb.  1,  1904. 

3.  Make  out  a  bill  for  the  following  articles  bought  dur- 
ing March  and  April.  Supply  the  names  of  buyer  and 
seller,  also  the  dates : 

23£  yards  of  silk,  at  80^;  If  yards  of  lace,  at  $2.40; 
64  yards  of  muslin,  at  6 J^ ;  8  spools  of  sewing  silk,  at  7^ ; 
4  pairs  of  stockings,  at  65$;  6  yards  of  linen,  at  87|^; 
-J-  dozen  collars,  at  $  2.10. 

4.  Make  out  a  bill  for  the  following  goods,  bought 
June  15: 

3  cases  of  torpedoes,  at  $  2.20 ;  12  boxes  of  firecrackers, 
at  $1.62-J-;  3  gross  pinwheels,  at  $1.35;  5  gross  sky- 
rockets, at  $  3.25  ;  2  dozen  balloons,  at  $  2.25 ;  45  lanterns, 
at  9£ 

Note.  —  The  date  is  written  only  at  the  top  of  the  bill  when  all  the 
articles  are  bought  at  one  time. 

SHORT  METHODS  — REVIEW. 

If  the  school  is  to  train  for  life,  it  must  accustom  pupils  to  use 
modes  of  calculation  followed  in  the  business  world. 

In  their  previous  work,  pupils  have  employed  $^  instead  of  25?, 
-$£  instead  of  12^?,  etc.  They  have,  for  instance,  found  the  cost  of 
32  pounds  at  25?  per  pound,  by  multiplying  $£  by  32.  While  the 
result  in  example  4  is  the  same,  25  pounds  at  32?,  the  analysis  is  dif- 
ferent.   The  following  is  suggested  : 

100  pounds  at  32?  would  cost  $  32,  \  of  100  pounds  would  cost  \  of 
432,  or  #8. 

The  rule  generally  given  for  the  multiplication  by  26  is  to  annex 


Review.  103 

two  ciphers  to  the  multiplicand  and  to  divide  by  4.  In  practice,  the 
ciphers  need  not  be  annexed  actually  or  mentally.  To  multiply  19  by 
25,  the  pupil  divides  19  by  4,  getting  4  for  quotient ;  to  this  he  adds  75 
for  the  3  remainder,  getting  475  for  the  result. 

174.  Oral  Exercises. 

1.  Multiply  by  25 : 

16,  19,  21,  23,  25,  29,  33,  36,  42,  48. 

2.  How  many  square  feet  in  a  lot  84  feet  long,  25  feet 
wide? 

3.  What  is  the  weight  of  25  barrels   of  flour,  each 
weighing  196  pounds? 

4.  Find  the  cost  of  25  pounds  of  coffee  at  32^  per 
pound. 

5.  What  will  a  woman  have  to  pay  for  25  yards  of  silk 
at  $1.60  per  yard? 

6.  A  man  sold  25  cows  at  $44  each.    How  much  did 
he  receive  for  them? 

7.  Multiply  64  by  12i 

8.  Find  the  cost  of  12^-  bushels  of  wheat  at  96^  per 
bushel. 

9.  At  $12.50  per  barrel,  how  much  should  I  pay  for 
56  barrels  of  pork? 

10.  How  many  pens  in  12^-  gross?     (144  to  gross.) 

11.  Find  the  cost  of  121  pounds  of  tea  at  56  jt  per  pound. 

12.  How  many  square  yards  in  a  field  96  yards  long,  75 
yards  wide? 

175.  Sight  Exercises, 

To  multiply  427  by  25  the  pupil  considers  4  as  the  divisor.  He 
writes  on  his  paper  1,  then  0,  then  6,  annexing  75  for  the  3  remaining. 
Ans.     10,675. 

Example  5 :  25  x  686  is  the  same  as  686  x  25. 

Example  9 :  To  multiply  by  250,  consider  three  ciphers  annexed  to 
the  multiplicand. 


104  Chapter  Two. 

Example  11 :  Divide  by  8,  annexing  two  ciphers  to  the  quotient 
when  there  is  no  remainder.  Annex  12£  when  the  remainder  is  1 ;  25, 
when  the  remainder  is  2 ;  etc. 

Example  19:  Consider  three  ciphers  annexed  and  divide  by  8. 

Write  only  the  answers : 


1. 

837  x     25 

8. 

25  x  2174 

15. 

12  J  x  1084 

2. 

763  x     25 

9. 

837  x    250 

16. 

12£  x  2196 

3. 

934  x     25 

10. 

763  x    250 

17. 

12£  x  3670 

4. 

508  x     25 

11. 

864  x    12£ 

18. 

\2\  x  6281 

5. 

25  x    686 

12. 

776  x    121- 

19. 

864    x    125 

6. 

25  x    301 

13. 

236  x    12£ 

20. 

776    x    125 

7. 

25  x  1039 

14. 

404  x    12\ 

21. 

125    X1020 

176.   Sight  Exercises. 

Pupils  do  much  unnecessary  work  in  rearranging  numbers  and  in 
writing  fractions  over  again  with  a  common  denominator.  A  few  of 
these  examples  should  be  written  on  the  blackboard  from  time  to  time, 
and  the  teacher  should  require  the  pupil  to  write  nothing  but  the 
answers. 


Add: 

1.   3£  +  5£ 

4.    11J  +  4J. 

7. 

8i  +  6J 

2.   4£  +  8f 

5.      7f  +  9^ 

8. 

15|  +  8J 

3.   9|  +  7f 

6.      5f  +  2| 

9. 

9f  +  5£ 

177.    Subtract  at 

sight : 

10.   23^~19| 

14.     9£-2* 

18. 

35J    -Si 

11.   16f   -    9f 

15.    lOJ-5^ 

19. 

11*   -6* 

12.   18f   -   S\ 

16.    14J-8A 

20. 

43A-8J 

13.    15f    -    Si  17.    27£-7|  21.    50J-    -  4f 


Review.  105 

178.   Multiply  at  sight,  18|  x  4. 

f  x  4  =  3.    4  eights  are  32,  and  3  are  35  (put  down  5).     4  ones 
are  4  and  3  are  7  (put  down  7).  Ans.  75. 

The  pupil  should  write  only  the  answers. 


22. 

27}  x  10 

27. 

15}  x  3 

32. 

37}  x  3 

23. 

33}  x  12 

28. 

13}  x  4 

33. 

45|x5 

24. 

16f  x8 

29. 

20}  x  11 

34. 

23}  x  4 

25. 

17|x8 

30. 

40f  x5 

35. 

17}  x  6 

26. 

19|x6 

31. 

16|x7 

179.  Divide  at  sight. 

"When  the  divisor  is  an  integer  less  than  12,  the  pupil  should  not 
reduce  the  mixed  number  in  the  dividend  to  an  improper  fraction, 
by  3,  the  pupil  first  gets  the  whole  number  of  the 

Ans.  89}. 

In  dividing  248£  by  4,  the  pupil  obtains  62  as  the  quotient  of  248 
by  4 ;  he  then  finds  \  of  \,  which  is  \.  Ans.  62} 

In  dividing  202£  by  5,  the  quotient  is  40,  and  the  remainder  is  2} 
which  is  reduced  to  f .    One-fifth  of  this  is  }  Ans.  40} 

In  dividing  183|  by  6,  the  quotient  is  30  with  a  remainder  of  3£, 

Ans.  30  J|. 

46.  7)97^ 

47.  10)87} 

48.  4)66} 

49.  3)94} 

50.  5)83} 

SIGHT  APPROXIMATIONS. 

180.  Give  approximate  answers  in  whole  numbers.    Solve 
for  the  exact  answers. 

1.  17  &  X  3fJ;  or,  about  17  x  about  4. 

2.  25^  -7- 1|;  or,  about  25  -r-  J  nearly. 


or^.    *of¥  =  lf 

36.     3)45f 

41. 

8)37} 

37.      4)56} 

42. 

9)48} 

38.    12)36} 

43. 

6)25} 

39.      5)72} 

44. 

7)10} 

40.    11)834- 

45. 

6)751 

io6 


Chapter  Two. 

3. 

61  x  H                           7. 

799f|-=-99H 

4. 

300^  +  llff                   8. 

7t%x7A 

5. 

86|xH                             *• 

TixllA 

6. 

35J  +  3H                         10. 

M*  x  f 

181.  Give  answers  in  whole  numbers : 

1.  5.75  x  9.999;  or,  5.75  x  10  nearly. 

2.  24.002  +  .4999 ;  or,  24  -?-  nearly  f 

3.  25.125x11.834  7.  799.9  x  .103 

4.  36.843  -j-  6.105  8.  7.999x7.999 

5.  86.4 -.983  9.  7.001x12.003 

6.  32.04x5.001  10.  64.001 -f- .249 

182.  Give  the  cost,  approximately,  of : 

1.  49  horses  at  $  199  each.     ($200  x  49.) 

2.  199  yd.  2  ft.  11  in.  of  cloth  at  $2.50  per  yard. 

3.  3  lb.  15  oz.  of  butter  at  25?  per  lb. 

4.  398  coats  at  $  12  each. 

5.  7  bu.  3  pk.  7  qt.  potatoes  at  $2  per  bushel. 

6.  798  base-balls  at  25  cents  each. 

7.  19  gal.  3  qt.  1  pt.  alcohol  at  $2.49  per  gallon. 

8.  995  lb.  tea  at  59f  cents  per  pound. 

9.  7  houses  at  $4995  each. 

10.   507  pounds  of  hay  at  99  cents  per  100  pounds. 

183.  Oral  Eeview  Exercises. 

1.  What  is  |  of  60  ?    fof35? 

2.  A  man  sold  a  boat  for  $  8,  which  was  \  of  what  it 
cost  him.     What  did  it  cost  him  ? 

3.  A  man  having  $35,  gave  away  -J-  of  it.     How  much 
had  he  left  ? 


Review.  107 

4.  How  many  inches  are  there  in  f  of  a  yard  ?    f  of  a 
yard  ?     -J  of  a  yard  ? 

5.  If  6  eggs  cost  12  cents,  what  will  5  dozen  cost  ? 

6.  How  much  is  f  less  \  ?    £  less  i  ? 

7.  Change  to  improper  fractions  :  7$,  9J,  6f,  2^,  6£. 

8.  How  many  apples  at  2^  apiece  are  worth  as  much  as 

4  peaches  at  5fi  apiece  ? 

9.  Which  is  the  greater  and  how  much :  f  of  $  24,  or  | 
of  I  25  ? 

10.  Change  to  mixed  numbers :  &£-,  A^,  -fj,  -4/,  -^. 

11.  There  are  45  pupils  in  school  and  -J  of  them  are  girls. 
How  many  are  boys  ? 

12.  Add  8J  and  7$.    5f  and  1\. 

13.  If  it  takes  5  men  15  days  to  do  a  piece  of  work,  how 
long  will  it  take  10  men  to  do  it  ? 

14.  What  will  2  bushels  of  corn  cost,  if  £  peck  costs  15 
cents  ? 

15.  If  it  costs  25  cents  to  set  one  shoe,  what  will  it  cost 
to  shoe  a  span  of  horses  all  around  ? 

16.  Bought  5  yards  of  ribbon  at  16^,  and  3  yards  of  linen 
at  75^,  and  gave  a  two-dollar  bill.     What  was  my  change  ? 

17.  If  7  yards  cost  84^,  how  many  yards  can  be  pur- 
chased for  $1? 

18.  If  6  oranges  cost  15^,  how  much  will  8  cost  ? 

19.  1-J-  pecks  of  peanuts  cost  $  0.48 ;  what  will  one  quart 
cost? 

20.  Two  boys  walked  in  opposite  directions ;  one  walked 

5  miles  an  hour,  the  other  4  miles  an  hour.     How  far  apart 
were  they  in  six  hours  ? 

21.  If  I  of  a  yard  of  cloth  cost  6^,  how  much  cloth  can 
be  bought  for  40?  ? 


108  Chapter  Two. 

22.  At  £  a  dollar  per  day  for  board,  how  many  days' 
board  can  you  get  for  $  7.50  ? 

23.  Charles  picked  £  peck  of  berries,  William  £  peck, 
and  Alfred  £  peck.     How  much  did  they  all  pick  ? 

24.  How  much  more  is  }  of  80^  than  f  of  75^  ? 

25.  A  boy  bought  3^  pounds  of  butter  for  his  mother. 
How  many  ounces  did  he  buy  ? 

26.  If  a  man  is  50  years  old  now,  how  old  was  he  22 
years  ago  ? 

27.  Mary  works  4  hours  and  40  minutes,  and  Nellie 
works  2  hours  and  20  minutes.  How  many  hours  do  they 
both  work  ? 

28.  If  you  should  receive  15  cents  at  one  time,  26  cents 
at  another  time,  and  14  cents  at  another  time,  how  much 
would  you  receive  in  all  ? 

29.  If  you  had  f  of  a  dollar,  and  should  buy  a  pound  of 
soda  for  8^  and  a  pound  of  tea  for  45^,  how  much  would 
you  have  left  ? 

30.  If  you  give  a  boy  $  10,  how  many  mills  do  you  give 
him? 

31.  50-12-9-19  = 

32.  72—7  times  9  =  what  number? 

33.  45  is  how  much  less  than  5  times  12? 

34.  (f  of  80) +  25  = 

35.  (35 +  15) -(14 -9)  = 

36.  $  =  how  many  sixths  ? 

37.  2\  —  how  many  fourths? 

38.  Give  the  exact  divisors  of  20.        40.   From  \  take  J. 

39.  Give  the  three  factors  of  30.  41.   2\  +  \  —  \  = 
42.   At  12  cents  a  dozen,  what  will  a  gross  of  buttons  cost? 


Review.  109 

43.  How  many  inch  cubes  will  exactly  cover  a  square 
foot  of  surface? 

44.  What  does  f  of  anything  mean  ? 

45.  1  gallon  2  quarts  and  1  gallon  1  quart  are  how  many 
quarts  ? 

46.  If  4  yards  of  muslin  cost  48  cents,  how  much  will 
one-third  of  a  yard  cost  ? 

47.  Paid  $4.86  for  6  bushels  of  rye.  What  was  the  price 
per  bushel  ? 

48.  Bought  3  dolls  at  49  cents  each.     Total  cost  ? 

49.  If  12  hats  cost  $  7,  what  will  be  paid  for  36  hats  ? 

50.  If  2  pounds  and  5  ounces  butter  cost  74  cents,  what 
will  be  the  cost  of  3  pounds  and  2  ounces  ? 

51.  How  many  bottles  holding  1\  pints  will  be  needed  to 
contain  2\  gallons  ? 

52.  A  bag  of  flour  contains  \  of  a  barrel  of  196  pounds. 
How  many  pounds  does  the  bag  contain  ? 

53.  What  will  be  the  cost  of  a  dozen  heads  of  cauliflower 
at  the  rate  of  2  for  25  cents  ? 

54.  Twenty  examples  are  given  out.  A  pupil  that  cor- 
rectly answers  all  receives  100  per  cent.  What  per  cent 
will  a  pupil  receive  that  solves  16  examples  ? 

55.  A  woman  receives  $40  interest  a  year.  How  much 
does  she  receive  in  3  years  and  6  months  ? 

56.  A  man  bought  some  cows  at  $  35  each,  and  the  same 
number  at  $  45  each.     What  was  the  average  price  ? 

57.  A  girl  received  100  credits  in  each  of  three  studies, 
and  80  credits  in  the  fourth.  What  was  the  total  number 
of  credits  in  the  four  studies  ?     What  was  her  average  ? 

58.  A  square  floor  contains  144  square  feet.  How  many 
feet  long  and  wide  is  it  ? 


no  Chapter  Two. 

59.  -J  yard  cloth  costs  $  f .    What  is  the  price  per  yard  ? 

Note.  —  In  dividing  one  fraction  by  another  mentally,  reduce  both 
to  their  common  denominator. 

|  price  of  a  yard  =  $  £.  ■&  price  of  a  yard  =  $  fo  Multiplying  by 
12,  8  times  price  of  a  yard  =  $  9. 

60.  A  man  owning  f  of  a  vessel  sells  f  of  his  share.  What 
part  of  the  vessel  does  he  then  own  ? 

61.  A  barrel  contains  196  pounds  of  flour;  the  barrel 
weighs  24  pounds.     What  is  the  weight  of  both? 

62.  A  family  uses  3J  pounds  of  sugar  per  day.  How  long 
will  24^-  pounds  last  ? 

63.  How  much  will  be  the  cost  of  3  pounds  of  25-cent  coffee 
and  1  pound  butter  at  36^  ? 

64.  If  |  of  a  pound  of  candy  costs  30^,  what  will  be  the 
cost  of  J  of  a  pound  ? 

Note.  — 6  eighths  cost  30^,  what  will  7  eighths  cost  ? 

65.  A  tailor  has  a  piece  of  cloth  containing  2\  yards;  he 
sells  If  yards.     What  part  of  the  piece  does  he  sell  ? 

66.  How  many  quarts  in  1  bushel  1  peck  and  1  quart? 

67.  Keduce  ff  to  lowest  terms. 

68.  24  J-  yards  of  cloth  are  used  for  7  coats.  How  many 
yards  in  each  coat  ? 

69.  If  cloves  are  worth  20^  per  \  pound,  how  much  will 
be  paid  for  7  ounces? 

70.  At  3  oranges  for  5^,  what  will  be  the  cost  of  1^ 
dozen  oranges? 

71.  My  purchase  amounts  to  $1.29.  I  give  the  store- 
keeper a  $2  bill.     How  much  change  do  I  receive? 

72.  A  bushel  of  nuts  was  sold  for  5^  per  quart.  How 
much  money  did  it  bring  ? 


Review.  ill 

73.  How  many  days  in  the  summer  months,  June,  July^, 
and  August  ? 

74.  John  had  40  cents.  After  earning  24  more,  he  spent 
his  money  for  marbles  at  4  cents  each.  How  many  did  he 
buy? 

75.  George  was  sent  to  the  store  with  50^.  He  bought 
6  pounds  of  rhubarb  at  2^  a  pound,  and  two  bunches  of 
radishes  at  5^  a  bunch.     How  much  money  had  he  left  ? 

76.  At  $  10  a  ton  what  will  be  the  cost  of  1000  pounds  ? 

77.  There  are  16  rooms  in  a  building  with  50  desks  in  a 
room.     How  many  desks  in  all  ? 

78.  Edgar  earned  $2.75  one  week,  and  $2.50  the  next 
week.     How  much  did  he  earn  in  both  weeks  ? 

79.  $  6  is  f  of  how  many  dollars  ? 

80.  Charles  began  work  at  2.45  p.m.  and  stopped  at 
5.15  p.m.     How  long  did  he  work  ? 

81.  29  +  18  +  30  +  9  +  8  +  7=? 

82.  ^  of  22  is  how  many  times  4  ? 

83.  Bought  a  horse  for  $45  and  a  saddle  for  $35,  and 
then  sold  them,  gaining  $20.  For  how  much  were  they 
sold? 

84.  Add  these  numbers :  12,  15,  9,  13,  11,  7,  and  24. 

85.  If  you  buy  6  yards  of  tape  at  7  cents  a  yard,  and 
4  yards  of  silk  at  7  dollars  a  yard,  what  will  you  give  for 
both  tape  and  silk  ? 

86.  Bought  8  firkins  of  butter  for  $72,  and  gave  2  of 
them  for  9  yards  of  cloth.  What  was  a  yard  of  the  cloth 
worth  ? 

87.  Mr.  Brown  mixed  3  pounds  of  black  tea  worth  40 
cents  a  pound  with  1  pound  of  60-cent  green  tea.  What  is 
the  mixed  tea  worth  a  pound  ? 


112  Chapter  Two. 

184.   Written  Eeview  Exercises. 

1.  In  6987  days  how  many  minutes? 

2.  Find  the  cost  of  1,588,000  pounds  of  coal  at  $5.98 
a  ton. 

3.  How  many  cords  of  wood,  at  $  7.85  a  cord,  can  be 
purchased  for  $  59,730.65  ? 

4.  Divide  $  3,245,530  by  468. 

5.  Bought  8  bushels  3  quarts  valuable  seed  at  seven 
dollars  and  eight  cents  a  quart.  How  much  did  the  seed 
cost? 

6.  What  is  the  cost  of  19  gallons  2  quarts  of  cologne  at 
90^  a  quart? 

7.  Divide  f  of  $  60,800  equally  among  75  persons. 

8.  Bought  a  house  for  $23,650,  and  land  for  $73,640. 
For  how  much  must  I  sell  them  to  gain  $  4500  ? 

9.  Find  the  greatest  common  divisor  of  45  and  135. 

10.  A  grocer  bought  7200  gallons  of  oil,  one-third  of  it 
leaked  out,  and  he  sold  the  remainder  at  25  cents  a  gallon. 
How  much  did  he  receive  for  it  ? 

11.  From  two  and  four-tenths  yards  take  .445  of  a  yard. 

12.  Add  the  numbers  from  490  to  505  (inclusive). 

13.  If  56  pounds  of  sugar  cost  $  3.08,  what  will  24  pounds 
cost? 

14.  If  42  gallons  3  quarts  1  pint  of  cream  cost  $  27.44, 
what  will  32  pints  cost  ? 

15.  A  man's  bill  at  a  provision  store  was  $  6.66.  He 
had  bought  two  pecks  of  peas  for  $0.54  and  some  beans  for 
$  0.36.  The  rest  of  the  bill  was  for  sirloin  steak  at  $  0.32 
per  pound.     How  many  pounds  of  meat  had  he  bought? 

16.  From  1,890,070  take  990,979. 

17.  If  a  train  travels  45  miles  per  hour,  how  far  will  it  go 
from  half-past  9  in  the  morning  to  a  quarter  of  7  in  the 
evening  ? 


Review.  113 

18.  A  mechanic  saved  $  35  per  month  for  11  months,  and 
$  20  the  twelfth  month.  His  expenses  averaged  $  3  each 
day  of  the  year.  What  were  his  daily  wages  for  the  300 
days  he  worked  ? 

19.  A  160-acre  farm  consists  of  5  fields.  The  first  con- 
tains 17.38  acres,  the  second  29.4  acres,  the  third  35.073 
acres,  the  fourth  25.875  acres.  How  many  acres  are  there 
in  the  fifth  field  ? 

20.  How  many  seconds  in  7  hours  15  minutes  ? 

21.  Find  the  total  cost  of  2  dozen  rockets  at  $7.50  per 
gross,  3  dozen  Roman  candles  at  $9.60  per  gross,  and  24 
dozen  pin  wheels  at  $  1.35  per  gross. 

(1  gross  =  12  dozen.) 

22.  Three  lots  of  tea  were  sold  for  $330.  The  second 
contained  twice  as  much  as  the  first,  and  the  third  three 
times  as  much  as  the  first.  The  third  lot  contained  330 
pounds.     Find  the  selling  price  of  the  tea  per  pound. 

23.  A  barrel  of  molasses  contained  40  gallons.  One- 
fourth  of  it  leaked  out.  If  the  molasses  cost  45  cents  per 
gallon,  what  price  must  be  charged  for  the  remainder  so 
that  there  will  be  no  loss? 

24.  If  12£  dozen  rockets  cost  $5.75,  what  will  15  dozen 
cost? 

25.  Show  by  drawings  that  -J  =  -j^,  and  that  f  =  |. 

26.  Write  the  first  five  prime  numbers  that  are  greater 
than  7. 

27.  Find  the  greatest  common  divisor  of  1220  and  2013. 

28.  Find  the  least  common  multiple  of  12,  15,  14,  6,  21, 
21,  and  24. 

29.  Find  the  prime  factors  of  1140. 

30.  Add  8fc  -£,  f  and  }  of  7*. 

31.  From  14^  pounds  of  butter,  5f  pounds  were  sold  to 
one  person  and  3£  to  another.     How  much  remained  ? 


H4  Chapter  Two. 

32.  A  man  bought  4  bushels  of  wheat  for  3f  dollars. 
What  fraction  of  a  dollar  did  one  bushel  cost? 

33.  If  f  of  a  bushel  of  oats  will  last  a  horse  one  day, 
how  long  will  4J-  bushels  last? 

34.  In  two  months  Ann  will  be  15  years  old.  How  old 
was  she  nine  months  ago  ? 

35.  A  boy  has  to  walk  from  his  home  to  a  house  If  miles 
east  of  his  home,  from  there  to  a  place  2\  miles  west  of  his 
home,  and  then  home.     How  far  has  he  to  walk  ? 

36.  I  lost  f  of  my  money,  then  found  f  of  what  I  had 
lost,  and  then  had  64  cents.     How  much  had  I  at  first  ? 

37.  Quotient  24J,  divisor  3^.  What  is  the  dividend? 
The  product  is  2|,  and  one  factor  is  -§.  What  is  the  other 
factor  ? 

38.  Bought  Z\  yards  of  muslin  at  7  cents  a  yard,  5^- 
yards  of  ribbon  at  3J  cents  a  yard,  and  2\  yards  of  cloth  at 
$  1.75  per  yard,  and  gave  a  ten-dollar  note  in  payment. 
How  much  change  did  I  receive? 

39.  Write  seven  million  nine  thousand  nineteen. 

40.  A  milliner  sells  3  pieces  of  ribbon  at  18  cents  per 
yard.  They  measure  4|  yards,  3f  yards,  and  5^-  yards 
respectively.     What  does  she  receive  for  the  ribbon  ? 

41.  How  many  feet  and  inches  in  T5^-  of  a  yard  ? 

42.  To  make  powder,  a  man  mixes  1\  pounds  of  saltpetre, 
IfV  pounds  of  sulphur,  and  as  much  charcoal  as  sulphur. 
How  many  pounds  of  powder  will  there  be  ? 

43.  Four  men  form  a  partnership ;  the  first  furnishes  -£■ 
of  the  capital,  the  second  -f,  and  the  third  -j^-.  What  frac- 
tion of  the  capital  is  furnished  by  the  fourth  ? 

44.  I  pay  15  cents  more  for  a  half-pound  of  tea  than  I 
pay  for  a  quarter-pound  of  the  same  tea.  What  is  its  price 
per  pound  ? 


Review.  115 

45.  After  doing  £  of  a  piece  of  work,  a  man  requires  3 
days  more  to  finish  it.  How  many  hours  does  he  take  to  do 
the  whole  work  if  he  works  8  hours  per  day  ? 

46.  If  1  pound  7  ounces  of  coffee  cost  46  cents,  what  will 
3  pounds  9  ounces  cost  ? 

47.  Add  6  hours  50  minutes  and  17  hours  10  minutes. 

48.  15  men  do  a  piece  of  work  in  lOf  days.  How  long 
would  it  take  5  men  to  do  the  same  work? 

49.  To  make  a  cloak,  3  yards  of  cloth  1^  yards  wide  are 
required.    How  much  cloth  £  yard  wide  would  be  required? 

50.  In  3  years  4  months  a  gas  company  manufactures 
4,200,000  cubic  feet  of  gas.  How  many  cubic  feet  are 
manufactured  per  year  ? 

51.  If  2|  dozen  hats  cost  $80,  what  will  be  the  cost  of  3 
hats? 

52.  A  boy  hires  a  boat  at  20  cents  per  hour.  How 
much  should  he  pay  if  he  uses  it  from  20  minutes  before  9 
a.m.  until  10  minutes  past  1  p.m.  ? 

53.  A  and  B  kill  an  ox.  A  takes  f  and  B  the  remainder. 
If  B's  share  weighs  361^  pounds,  what  is  the  weight  of  the 
ox? 

54.  A  grocer  buys  30  dozen  eggs  at  18  cents  per  dozen. 
He  sells  them  at  the  rate  of  15  eggs  for  25  cents.  What  is 
his  profit  ? 

55.  How  many  cents  in  T5g-  of  a  dollar  ? 

56.  What  fraction  of  18f  is  6f  ?     ' 

Suggestion.  —  What  fraction  of  18  is  6  ?    Which  is  the  divisor  ? 

57.  A  farmer  buys  a  horse  for  $140,  and  sells  it  at  an 
advance  of  -fa  of  the  cost.     What  is  the  selling  price  ? 

58.  In  1903,  A  was  36  years  old  and  B  was  If  times 
as  old.     In  1894,  B  was  how  many  times  as  old  as  A  ? 


n6  Chapter  Two. 

59.  From  the  sum  of  18^  and  25£  take  their  difference. 

60.  If  2f  acres  of  land  cost  $  220,  what  will  be  the  cost 
of  17 $  acres? 

Note.  —  Indicate  the  operations,  and  cancel. 

61.  A  can  do  a  piece  of  work  in  6  days,  B  can  do  it  in  6 
days,  C  can  do  it  in  6  days.  How  long  will  it  take  all 
three  working  together  ? 

62.  Find  the  value  of  1  f  t£*M. 

63.  A  man  sold  a  horse  for  }  of  its  cost,  losing  $40. 
What  did  the  horse  cost  him  ? 

64.  I  have  an  oblong  piece  of  land  which  is  96  feet  long 
and  72  feet  wide.  There  are  three  gateways;  one  is  two 
feet  wide,  one  is  three  feet  wide,  and  the  other  is  four  feet 
wide.  How  many  feet  of  fence  will  it  take  to  go  around 
the  field? 

65.  Add:  $83.34;  $67.58;  $50.37;  $62.50;  $35.75; 
$62.50;  $35.75;  $63.81;  $67.59;  $86.37;  $37.50; 
$15.09;  $57.32;  $49.63. 

66.  A  boy  bought  a  suit  of  clothes  for  $21,  boots  for 
$3.50,  overcoat  for  $15,  and  gloves  for  50  £  Paid  for 
these  things  in  work  at  $1.25  per  day.  How  many  days 
did  he  work? 

67.  If  $36.53  will  buy  6£  yards  of  cloth,  how  much  will 
■J-  yard  cost  ? 

68.  If  two  quarts  of  peaches  cost  25^,  what  will  half  a 
bushel  cost? 

69.  How  many  geographies  at  90^  apiece  can  be  bought 
for  $54? 

70.  Find  the  least  common  multiple  of  6,  24,  32,  48,  96. 

71.  Add:  87.5;  7004.3;  500.004;  21,090;  5040.29. 


Review.  117 

72.  Spent  $290  for  horses,  $286.75  for  carriages, 
$150.80  for  harness,  and  $12.75  for  blankets.  Gave  4 
fifty-dollar  bills  and  2  one-hundred-dollar  bills.  What  did 
I  still  owe  ? 

73.  How  many  bushels  of  oats  will  a  span  of  horses  eat 
in  4  weeks,  if  they  eat  24  quarts  a  day? 

74.  How  many  bottles,  each  holding  \  pint,  will  it  take 
to  hold  725  gallons  and  2  quarts  of  oil  ? 

75.  How  many  pounds  of  rice  at  12^  a  pound,  will  pay 
for  4  bushels  2  pecks  of  nuts  at  8^  a  pint  ? 

76.  A  man  had  $  600.  He  bought  a  horse  for  $  225,  a 
carriage  for  $  190.12,  and  a  harness  for  $  40.76.  He  then 
gave  away  £  of  what  he  had  left.     What  did  he  still  have? 

77.  Find  the  greatest  common  divisor  of  18,  24,  36. 

78.  The  least  common  multiple  of  12,  20,  and  30. 

79.  Find  the  cost  of  18,756  feet,  of  lumber  at  $30  per 
1000  ft. 

80.  A  field  is  14.25  rods  long  by  7.4  rods  wide.  What  is- 
its  area  in  square  rods  ? 

81.  A  rod  is  16.5  feet;  how  many  feet  are  there  in  24 
rods  ?     How  many  rods  are  there  in  231  feet  ? 

82.  How  many  marks  are  there  in  $100?  (A  mark  is- 
equal  to  23.8  cents.) 

83.  Add  3  and  4  tenths,  96  thousandths,  100  and  5  thou- 
sandths, 27  hundredths. 

84.  From  2700  take  27  hundredths. 

85.  Multiply  8  and  4  tenths  by  9  and  25  hundredths. 

86.  Divide  96  and  75  hundredths  by  322  and  5  tenths. 

87.  A  load  of  hay,  at  75  cents  per  100  pounds,  cost 
$13.98.     What  was  the  weight  of  the  hay  ? 


xi8 


Chapter  Two. 


88.  The  circumference  of  a  circle  is  3.1416  times  the 
diameter.  How  many  inches  in  the  circumference  of  a 
circle  whose  diameter  is  20  inches  ? 

89.  Show  by  a  diagram  the  number  of  pieces  of  wire  # 
yard  long  that  can  be  made  from  4  yards  of  wire. 

90.  Show  by  a  diagram  that  three-fourths  of  1  is  equal 
to  one-fourth  of  3. 

91.  If  two-thirds  of  a  yard  of  material  will  make  an 
apron,  how  many  aprons  can  be  made  from  two  yards  ?  Show 
by  a  diagram. 

92.  A  boy  paid  6  cents  for  three-eighths  of  a  pie.  What 
would  be  the  cost  of  the  whole  pie  at  the  same  rate? 
Make  a  drawing. 


93.  Seven-eighths  of  an 
acre  of  land  is  sold  for 
$140.  What  is  the  price 
of  an  acre? 


CHAPTER  III. 

PAGES 

Decimals 119  to  132 

Notation  and  Numeration,  Reduction,  Addition,  Sub- 
traction, Multiplication,  Division. 

United  States  Monet 132  to  133 

Denominate  Numbers 133  to  139 

Reduction,    Addition,    Subtraction,    Multiplication, 
Division  (two  denominations). 

Measurements 139  to  144 

Areas  of  Rectangles,  Areas  of  Right-angled  Triangles. 

Bills 144  to  145 

Percentage 145  to  147 

Interest 148  to  152 

Review  of  Simple  Numbers  and  Fractions  .        .     152  to  162 

Sight  Approximations,  Special  Drills,  Cancellation, 
Ratio,  Short  Methods,  Review  Fractions. 

Miscellaneous  Problems 162  to  172 

Oral  and  Written. 

DECIMALS. 

185.   Preliminary  Exercises, 

1.  Write   seven  tenths  as   a  common   fraction.      As  a 
decimal. 

2.  Write  three  hundredths  as  a  common  fraction.     As  a 
decimal. 

3.  Write  thirty-one  thousandths  as  a  common  fraction. 
As  a  decimal. 

119 


ISO  Chapter  Three. 

4.  Kead  the  following : 

.3  .09  .043 

.17  .007  .241 

5.  Write  each  of  the  foregoing  decimals  as  a  common 
fraction. 

186.  Notation  and  Numeration  of  Decimals. 

1.  7  tenths,  or  T77,  is  written  .7. 

2.  3  hundredths,  or  -j-J^,  is  written  .03. 

3.  53  hundredths,  or  ^j-,  is  written  .53. 

4.  9  thousandths,  or  TirVu">  *s  written  .009. 

6.  19  thousandths,  or  j^fo,  is  written  .019. 

6.  419  thousandths,  or  ^j^,  is  written  .419. 

7.  67  ten-thousandths,  or  l060700,  is  written  .0067. 

8.  1031  hundred-thousandths,  or  y^nHBiF'  *s  written  .01031. 

Note.  —  In  the  foregoing  examples,  it  will  be  observed  that  the  num- 
ber of  places  to  the  right  of  the  decimal  point  is  equal  to  the  number 
of  ciphers  in  the  denominator  of  the  corresponding  common  fraction. 

187.  Write  the  following  as  decimals : 

1.   314  ten-thousandths. 

Since  y^^  has  a  denominator  containing  four  ciphers,  the  decimal 
roust  have  four  places ;  a  decimal  cipher  must  be  written  after  the 
decimal  point.  Ans.  .0314. 

To  write  a  decimal,  write  the  numerator,  and  from  the  right, 
point  off  as  many  decimal  places  as  there  are  ciphers  in  the 
denominator,  prefixing  decimal  ciphers,  if  necessary. 

Note.  — Ciphers  between  the  decimal  point  and  the  first  significant 
figure  of  the  numerator  are  called  decimal  ciphers. 


Decimals.  121 

2.  217  hundred-thousandths. 

3.  83  hundredths. 

4.  7  millionths. 

5.  345  thousandths. 

6.  27  ten-thousandths. 

7.  325  and  7  thousandths. 

Ans.  325.007.  This  is  called  a  mixed  decimal,  which  consists  of  an 
integer  and  a  decimal. 

188.  The  word  and  is  used  in  reading  mixed  numbers  or 
mixed  decimals  to  separate  the  integer  from  the  common 
fraction  or  the  decimal. 

8.  42  and  56  hundred-thousandths. 

9.  150  and  62  millionths. 
10.   489  and  3  hundredths. 

189.  Eead  the  following: 

1.  .0346. 

Since  there  are  four  decimal  places,  the  denominator  is  1  with  four 
ciphers,  10000.  Ans.  346  ten-thousandths. 

2.  654.15  6.  25.006347 

3.  .000209  7.  3.259 

4.  60.0207  8.  .002468 

5.  684.007  9.  200.0035 

200.0035  read  as  200  and  35  ten-thousandths  might  be  mistaken  for 
235  ten-thousandths.  It  should  be  read  200  units  and  35  ten-thou- 
sandths, or  200  whole  number  and  35  ten-thousandths. 

10.  1000.0006  12.   2300.00021 

11.  300.075  13.  400.000007 


122 


Chapter  Three. 


190.   Changing  Common  Fractions  to  Decimals. 
Eeduce  -£%  to  a  decimal. 

■fc  means  3  -4-  32.  Performing  the  indicated  division, 
we  obtain  the  quotient  .09375.    -fa  =  .09375,  Ans. 

Divide  the  numerator,  ivith  the  necessary  ciphers 
annexed,  by  the  denominator.  The  number  of 
decimal  places  in  the  quotient  will  be  equal  to  the 
number  in  the  dividend. 


.09375 


Eeduce  to  decimals 


1. 

"T017 

2. 

A 

3. 

A 

4. 

H 

5. 

H 

6. 

A 

7. 

sort 

8-    * 


y. 

4000 

10. 

2  0  00 

11. 

H 

12. 

TT5" 

13. 

¥ 

14. 

« 

32)3.00000 
288 

120 
96 

240 
224 

160 
160 

15. 

250" 

16. 

F2T 

17. 

W 

18. 

T% 

19. 

T5~8" 

20. 

"ST"? 

21-   rrfcr 

191.  Changing  Decimals  to  Common  Fractions. 
What  is  the  denominator  of  a  decimal  fraction  ? 

What  prime  numbers  are  contained  in  10  ?  What  are  the 
only  factors  of  10  ?     The  prime  factors  of  100  ?     Of  1000  ? 

Can  yoVg  he  reduced  to  lower  terms  ?  Why  ?  Can  jfo 
be  reduced  to  lower  terms  ?  Why  ?  Can  y^jj^rr  be  reduced 
to  lower  terms  ?  How  can  we  tell  by  merely  looking  at  a 
decimal  whether  or  not  it  can  be  reduced  to  a  common 
fraction  of  lower  terms  ? 

192.  Written  Exercises. 

.  Reduce  the  following  to  common  fractions  —  lowest  terms. 
Do  not  find  the  greatest  common  divisor. 

1.    Reduce  .0064  to  a  common  fraction  —  lowest  terms. 

•0064  =  T&V*  =  rib  =  vfc,  -4ns. 


Decimals.  123 


2*   Reduce  .039  to  a  common  fraction. 

•039  =  xfjyi  Ans. 

This  cannot  be  reduced  to  lower  terms,  since  39  is  not  divisible  by  2 
or  5. 

3.   Reduce  .900  to  a  common  fraction  —  lowest  terms. 

tW(5  =  tVff  =  tV  Ans. 

Omit  the  decimal  point.  Write  in  the  form  of  a  common 
fraction,  and  reduce  to  lowest  terms. 

Ciphers  at  the  right  of  a  decimal  cancel  ciphers  in  the  denominator ; 
they  do  not,  therefore,  affect  the  value  of  the  decimal,  and  they  should 
be  omitted. 

193.   Reduce  to  common  fractions : 


1. 

.0075 

8. 

.37500 

15. 

.0009 

2. 

.36 

9. 

.144 

16. 

.816 

3. 

.0275 

10. 

.0006 

17. 

.15625 

4. 

.44 

11. 

.27 

18. 

.0375 

5. 

.03125 

12. 

.027 

19. 

.00625 

6. 

.486 

13. 

.00365 

20. 

.096 

7. 

.3750 

14. 

.96 

21. 

.326 

ADDITION  OF  DECIMALS. 

194.   Add  the  following,  reducing  the  common  fractions 
to  decimals. 

1.   18£  +  9.084  +  25^  +  163  +  2.09  +  .0975 

18.75 
9.084 
Write  the  decimals  so  that  tenths,  hundredths,  etc.,         25.05 
stand  in  the  same  column,  etc„  163. 

2.09 
.0975 


124  Chapter  Three. 

Write  the  numbers  so  that  decimal  points  stand  in  a  column. 
Add  as  in  integers,  and  place  the  point  in  the  sum  directly 
under  the  points  above. 

2.  275^  +  58.64  4  8.6796  4  30J-  4  8f  4  99 

3.  841^  +  93T^4-3Ii^  +  TlHir  +  684.1  +  i 

4.  250  +  1875.93  4-  Iff  +  A  +  608.94  +  .0005 

5.  8.6796  +  96.8  +  18f  +  34^  +  1876 

6.  40^  +  7.2832  4  86.3  4  128.46  4  2^ 

7.  540  41.32  4-576  4  1^  +  68^  4  395£ 

8.  5.308  4  .25  4  567.8  +  8.4825  4  49.795  +  8^ 

9.  7.08  +  23.04  4  8^  4  .348  +  3^  4.  7.00019 
10-  8^^4-8^4  507  +  28^4  6.8819 

SUBTRACTION  OF  DECIMALS. 

195.    Give  answers  in  decimals : 

1.      275.3 -87  A 

275.3 

Arrange  the  decimals  as  in  addition,  tenths  under       ^~  ^ . 

tenths,  etc.  ^nn\„    . 

188.26  Ans. 

Write  the  numbers  so  that  the  decimal  point  of  the  subtra- 
hend is  directly  under  the  decimal  point  of  the  minuend;  sub- 
tract as  in  integers,  and  place  the  point  in  the  remainder 
directly  under  the  points  above. 


2. 

387f  -  99.0127 

7. 

2345-345^- 

3. 

woo -T^n 

8. 

168^-54.8759 

4. 

62.365-48| 

9. 

618.42  -;576J 

5. 

198}- 13.6431 

10. 

1847H-344rfo 

6. 

24A-9rtt* 

11. 

622.5-6.243 

Decimals,  125 

196.  Oral  Problems. 

1.  Reduce  -fa  to  a  decimal. 

2.  Express  the  decimal  .3  J  as  a  simple  fraction. 

3.  What  decimal  of  a  ton  is  125  pounds  ? 

4.  One  hundred  fifty  marbles  are  divided  among  a  certain 
number  of  boys.  Each  receives  12  and  there  are  6  remaining. 
How  many  boys  are  there  ? 

5.  Express  the  decimal  .62^  as  a  simple  fraction. 

6.  What  decimal  of  a  peck  is  7  quarts  ? 

7.  If  8  men  can  do  a  piece  of  work  in  6  days,  in  how 
many  days  can  4  men  do  it  ? 

8.  If  Maria  spends  $  .75  a  day,  in  how  many  days  will 
she  spend  $9? 

9.  If  you  had  3J  oranges  to  divide  among  your  friends, 
giving  each  \  of  an  orange,  to  how  many  friends  would  you 
give? 

10.  -J-  of  14  is  -J-  of  what  number  ? 

11.  Change  .75  yards  to  feet  and  inches. 

12.  At  16|^  a  yard,  what  will  12  yards  of  ribbon  cost  ? 

13.  At  80^  a  pound,  what  do  4  ounces  of  tea  cost  ? 

14.  If  I  have  12  yards  of  ribbon,  to  how  many  girls  can  I 
give  f  of  a  yard  each  ? 

15.  A  boy  lives  10^-  rods  from  his  school.     How  far  does 
he  walk  in  a  day  to  attend  two  sessions  of  school  ? 

197.  Written  Problems. 

1.  In  the  written  number  54,372,  the  value  expressed  by 
the  5  is  how  many  times  the  value  expressed  by  the  2  ? 

2.  Find  the  sum  of  two  and  twenty-five  thousandths, 
five  and  twenty-seven  ten-thousandths,  forty-seven  and  one 
hundred  twenty-six  millionths,  one  hundred  fifty  and  seven 
ten-millionths. 


126  Chapter  Three. 

3.  In  a  mass  of  alloy  weighing  291.42685  pounds,  there 
were  found  40.0921  pounds  of  silver,  160.09090  pounds  of 
copper,  22.002  pounds  of  iron,  and  .426900  pounds  of  zinc. 
The  remainder  was  lead.    What  was  the  weight  of  the  lead  ? 

4.  How  many  bushels  of  oats  at  f  of  a  dollar  a  bushel 
will  pay  for  f  of  a  barrel  of  flour  at  $  5.40  a  barrel  ? 

5.  Add  3.684;  19.5;  .00875;  15,863.625;  8.7;  and 
100.4875. 

6.  Change  to  a  common  fraction  in  its  lowest  terms 
.009375.     Change  -fe  to  a  decimal. 

7.  If  f  pound  of  tea  costs  $■$■,  how  many  pounds  can  be 
bought  for  $7.50? 

8.  Change  to  common  fractions  .0075  and  .625. 

9.  Change  to  decimals  ^-,  •2-9-,  and  5 J,  and  add  the  results. 

10.  Reduce  to  common  fractions,  and  then  find  the  sum 
of  the  common  fractions:  .12^-,  .3^-,  .16f. 

1 1 .  Add  three  hundred  seventy-six  ten-thousandths,  forty- 
five  hundred-thousandths,  five  hundred  sixty-eight  thou- 
sandths, fourteen  and  fifteen  hundredths. 

12.  At  24  cents  per  gallon,  what  will  be  the  cost  of  16 
gal.  3  qt.  of  milk  ? 

MULTIPLICATION   OF  DECIMALS. 

198.    Give  answers  in  decimals : 

1.   Multiply  .000486  by  29.5. 

Place  the  units'  figure  (9)  of  the  multiplier  under  the  last  figure  (6) 
of  the  multiplicand.    486  millionths  multiplied  by  2  tens  gives  a  prod- 
uct of  972  hundred-thousandths,  or  .00972  ;  place  the 
right-hand  figure  (2)  of  this  product  under  the  2  of  the 

multiplier,  etc.  — 

00Q79 
The  result,  .0143370,  contains  seven  decimal  places,     -w^i* 

which  is  equal  to  the  six  in  the  multiplicand  plus  the  one  ^374 

in  the  multiplier.     Rejecting  the  unnecessary  cipher  at  2430 

the  right,  the  product  is  .014337,  Am.  .0143370 


Decimals. 

2.   Multiply  29.5  by  .000486. 

The  units'  figure  of  the  multiplier  may  be 
considered  as  zero. 

Ans.     .014337. 


127 

29.5 

0.000486 
.01180 
2360 
1770 


.0143370 


Multiply  as  in  whole  numbers,  and  from  the  right  of  the 
product  point  off  as  many  decimal  places  as  there  are  decimal 
places  in  both  factors. 


Multiply : 

1.     24.75  x  3.02 

6. 

1.876  x  34 

0 

2.        98|  x  .00046 

7. 

3.48  x  4.8665 

3.    148^x12.5 

8. 

.43J  x  1A 

4.      380^-  x  .012 

9. 

192.38  x  .238 

5.    .09375x1.48 

10. 

26.4  x  .016 

DIVISION  OF  DECIMALS. 


199.    1.   Divide  7.345  by  .29. 


Make  the  divisor  a  whole  number  by  mov- 
ing the  decimal  point  two  places  to  the  right, 
which  multiplies  the  divisor  by  100  ;  and  make 
a  corresponding  change  in  the  dividend.  Di- 
viding 734.5  by  29  gives  a  quotient  of  25.3275+. 
Since  the  quotient  is  to  be  limited  to  three 
decimal  places,  8  followed  by  a  minus  sign  is 
substituted  for  the  7,  to  indicate  that  the 
fourth  decimal  figure  is  at  least  5. 


25.327 

/29)7/34.500 

58_ 

154 

145 

95 

5L 
80 
58_ 
220 
203 
17 


Ans.    25.328- 


i*8 


Chapter  Three. 


2.   Divide  753  by  4.18. 


Kemoving  the  decimal  point  in  the  divisor 
two  places  to  the  right  multiplies  the  divisor 
by  100.     Annex  two  ciphers  to  the  dividend. 

As  the  fourth  decimal  figure  in  the  quo- 
tient is  greater  than  5,  the  3  is  changed  to  a 
4,  followed  by  a  minus  sign. 

Arts.     180.144  - 


3.   Divide  .8756  by  4326. 

The  decimal  point  in  the  quotient  is  placed 
over  the  new  decimal  point  in  the  dividend, 
the  necessary  decimal  ciphers  being  sup- 
plied. A  +  sign  is  placed  after  the  last 
quotient  figure  to  show  that  the  next  quo- 
tient figure  is  less  than  5. 


180.143 

4/18.)753/00.000 
418 

3350 
3344 


60.0 
41.8 
18.20 
16  72 
1480 
1254 
226 
Arts.    .0002024  + 

4326).8756000 
8652 
10400 
8652 


17480 

17304 

176 


Make  the  divisor  a  whole  number  by  removing  the  decimal 
point,  and  make  a  corresponding  change  in  the  dividend. 
The  number  of  decimal  places  in  the  quotient  will  be  equal  to 
the  number  in  the  dividend  as  changed. 


200.   Written  Exercises. 

Divide : 

1.     4.054 -j- 18.25 

10. 

62.478  +  4279 

2.     123.5 -j- 384 

11. 

346.25  -f-  64.8 

3.        471  -*-  5.325 

12. 

9.1342  -4-  208.3 

4.      .3126  -h  .0134 

13. 

1784  -f-  29.57 

6.   12.345 -r- .0047 

14. 

343.71  -f- 1.127 

6.     .8756  -*-  4.322 

15. 

83.087  -5-  5.37 

7.            8  -f- 122 

16. 

137.84  -5-  7.91 

8.     .3678  -*-  .9125 

17. 

38.9008  -*-  .523 

9.     48.45  -*-  .089 

18. 

.81074  h-  .0091 

Decimalso 


129 


201.    Solve  by  short  division: 

1.   Divide  18.756  by  3000. 

Cancel  the  ciphers  in  the  divisor,  thereby 
dividing  it  by  1000.  Move  the  decimal  point 
in  the  dividend  three  places  to  the  left,  which 
divides  it  by  1000.  Place  the  decimal  point 
in  the  quotient  under  the  new  decimal  point 
in  the  dividend. 


3000).O18/756 

.006252  Ans. 


2. 

48.36-5-4000 

11. 

48.64-5-200 

3. 

.4824-5-12000 

12. 

.00531-5-90000 

4. 

11.011-5-700 

13. 

96.51-5-60 

5. 

3.6504-90 

14. 

87.5^-500 

6. 

45.63-5-1500 

15. 

183.275-5-10000 

7. 

130.13-5-1100 

16. 

1.7632-5-1600 

8. 

.8712-5-60 

17. 

1.5639-5-130 

9. 

3.075-5-5000 

18. 

614.4-5-120 

10. 

.07056-5-140 

19. 

.8008-7000 

202.   Perform  indicated  operations. 

Change  the  divisor  to  a  whole    number,   making  corresponding 
change  in  the  dividend.     Cancel. 


2. 


7 
34.2  x/OT 

tm 

.249  x  3.92 
.098 

.083  x  .72 
288 

.6876  x  .27 
.081 


234       .001 

239.4  5.    ffi/ffx-#3--.234 

I&0 


6. 


7. 


3.1416  x  2.3 

.7854 

7.72  x  65 
19.3 

450  x  23.8 
1.19 


130  Chapter  Three. 

34.3  x  8.1  2.75  x  .801 

*    .49x100  '      1.1x6 

.576  x  6.3  .306  x  8.75 

'   14.4x25  '       .9x68 

203.  Eeduce  to  common  fractions  —  lowest  terms. 

1.  Eeduce  .3^  to  a  common  fraction  —  lowest  terms. 

.3£  is  a  complex  decimal ;  that  is,  a  decimal  and  a  common  fraction 
written  together.  It  may  be  written  as  the  complex  fraction  ^3, 
which  means  S\  -h  10. 

Note.  —  A  complex  fraction  is  one  which  has  a  fraction  in  the 
numerator  or  in  the  denominator  or  in  both. 

2.  Eeduce  .006 \  to  a  common  fraction  —  lowest  terms. 

.006*  =  .  00625  =  T^b; 
dividing  both  terms  by  25,  we  get  jffo  ; 
dividing  both  terms  by  25,  we  get  T^,  Ans. 

3.  .33J  6.    .01£  9.    .04£ 

4.  .16|  7.    .06|  10.   .76ff 

5.  .142^  8.   .833|  11.   .037£ 

204.  Oral  Exercises. 

1.  Divide  6  by  .03. 

2.  f  is  what  part  of  2  ? 

3.  What  is  the  product  of  one  hundred  by  one-hundredth? 

4.  Subtract  25  thousandths  from  5. 

5.  What  will  150  pounds  of  coffee  cost  at  the  rate  of  3 
pounds  for  50  cents  ? 

6.  What  will  be  the  cost  of  3  pecks  of  cherries  at  2 
cents  a  pint? 


Decimals.  131 

7.  Divide  -§  by  f . 

8.  At  3  oranges  for  5  cents,  what  will  be  the  cost  of  4 
dozen  oranges  ? 

9.  If  a  man  walks  ^  of  a  mile  in  10  minutes,  bow  far 
can  be  walk  in  an  hour  and  a  half  ? 

10.  A  woman  bought  12  yards  of  cloth  at  70^  a  yard ; 
she  paid  $  5  in  cash,  and  the  rest  in  butter  at  20^  a  pound, 
How  many  pounds  of  butter  did  she  give  ? 

205.   Written  Exercises. 

1.  Divide  the  sum  of  .736  and  1.2854  by  their  difference= 

2.  Divide  .1  by  .2,  and  .35  by  35,  and  find  the  product 
of  the  quotients. 

3.  Eeduce  -^fa  to  a  decimal,  and  divide  it  by  .3125. 

4.  Divide  .12096  by  .032. 

5.  Multiply  .00273  by  3000.456,  and  divide  the  .product 

by  .08. 

6.  Divide  12.3125  by  .000625. 

7.  Divide  51.5  by  412,  and  412  by  51.5. 

8.  Multiply  31.5  by  27.9,  and  divide  the  product  by 
9.765. 

4  9t 

9.  Eeduce  ^=^-. 

3-H 

10.  Find  the  value  of  '0Q21  *  3004. 

.024 

11.  What  will  be  the  duty  on  175  kilograms  of  wool  at 
33  cents  per  pound  ?     (1  kilogram  =  2.2046  pounds.) 

12.  How  much  is  the  fraction  f  increased  or  diminished 
when  2  is  added  to  each  of  its  terms  (numerator  and  denom- 
inator) ? 


132  Chapter  Three. 

13.  Find  the  cost  of  360  meters  of  cloth  at  $1.10  per 
yard  (1  meter  =  39.37  inches). 

14.  Find  the  cost  in  United  States  money  of  386  hats  at 
24  francs  each  (1  franc  =  19.3  cents). 

15.  Find  the  cost  in  United  States  money  of  480  meters 
of  cloth  at  1.10  marks  per  meter  (1  mark  =  23.8  cents). 

16.  A  merchant  bought  30  pieces  of  cloth,  each  contain- 
ing 41.6  yards,  for  $3,875  per  yard,  and  25  pieces  of  36.8 
yards  each,  for  $  4.125  per  yard.  He  sold  the  entire  lot  for 
$  3.96  per  yard.     How  much  did  he  gain  or  lose  ? 

17.  An  importer  received  a  box  of  chemicals  weighing 
122  grams,  each  gram  containing  15.432  English  grains,  on 
which  he  paid  a  duty  of  $.05  per  grain.  What  was  the 
amount  of  duty  ? 

18.  A  dealer  exported  374.319  bushels  of  corn,  receiving 
in  exchange  coal  at  the  rate  of  1  ton  of  coal  for  15.124 
bushels  of  corn.      How  much  coal  did  he  receive? 

19.  .75  is  what  part  of  3.25  ? 

20.  Keduce  .005025  to  a  common  fraction. 

UNITED  STATES  MONEY. 
206.   Written  Exercises. 

1.  Find  the  cost  of  24,400  bricks  @  $  6.25  per  M. 

M  means  1000.  24,400  =  24.4  M.  Since  the  cost  per  thousand  is 
$6.25,  24.4  thousand  will  cost  24.4  times  $6.25. 

2.  760  pounds  of  hay  @  95  cents  per  cwt.  (100  pounds) 

($.95x7.6) 

3.  48,600  laths  @  $  2.80  per  M. 

4.  39,250  stamped  envelopes  @  $21.30  per  thousand. 
6.   1875  pounds  of  straw  @  68  cents  per  cwt. 


Denominate  Numbers.  133 

6.  108,745  Philadelphia  bricks  @  $22  per  M. 

7.  14,860  oranges  @  75^  per  hundred. 

8.  2376  eggs  @  13J^  per  dozen. 

9.  4500  cigars®  $35  per  M. 

10.  28  dozen  wax  candles  @  $13.50  per  gross  (144). 

Solve  by  cancellation  where  possible  : 

1 1 .  38,648  pounds  of  wheat  @  90  f  per  bushel  (60  pounds). ' 

Since    there    are    60    pounds     in    a    bushel,    38,648    pounds  = 

8.09 

^^  bushels.     At  90  cents  per  bushel,  the  cost  is  $'^x38648,  etc. 
60  60 

Note.  —  In  cancelling,  be  careful  not  to  strike  out  a  cipher  in  60 
and  one  in  .90,  without  inserting  a  decimal  cipher. 

12.  18,964  pounds  of  coal  @  $5  per  ton  (2000  pounds). 

13.  48,576  pounds  of  oats  @  36/  per  bushel  (32  pounds). 

14.  69,104  pounds  of  rye  @  91-J-^  per  bushel  (56  pounds). 

15.  74,816  pounds  of  corn  @  48^  per  bushel  (56  pounds). 

DENOMINATE  NUMBERS. 
207.   Written  Exercises. 

1.    Change  12  pounds  and  9  ounces  to  ounces. 
Since  there  are  16  ounces  in  1  pound,  in  12 


16  oz. 


pounds  there  are  12  times  16  ounces,  or  192  ounces. 

i 
In  12  pounds  9  ounces,  there  are  192  ounces  +  9 

ounces,  or  201  ounces.  \ f  *D-     jj  oz" 

The  work  may  be  arranged  in  this  way.  Above  201  oz. 

the  ounces,  write  the  number  of  ounces  in  a  pound, 

viz.  16.     Multiply  16  ounces  by  12,  adding  in  the  ^ns'  201  oz. 

9  ounces. 


134  Chapter  Three. 

Change : 

1.  20  rods  and  3  yards  to  yards. 

2.  2  miles  to  yards. 

3.  3  days  and  17  hours  to  hours. 

4.  24  minutes  and  15  seconds  to  seconds, 

5.  8  tons  and  1675  pounds  to  pounds. 

6.  43  gallons  and  8  quarts  to  quarts. 

7.  75  gallons  to  pints. 

8.  19  bushels  and  3  pecks  to  pecks. 

9.  .03125  ton  to  pounds  and  ounces. 
10.  -J  yard  to  feet  and  inches. 

208.   "Written  Exercises. 
Change : 

1.  975  ounces  to  pounds  and  ounces. 

2.  396  inches  to  yards. 

3.  517  hours  to  days  and  hours. 

4.  1694  seconds  to  minutes  and  seconds. 

5.  9314  pounds  to  tons  and  pounds. 

6.  987  pints  to  gallons,  quarts,  and  pints. 

7.  1485  quarts  to  pecks  and  quarts. 

8.  185  pecks  to  bushels  and  pecks. 

9.  840  hours  to  weeks. 

10.  12  hours  to  the  fraction  of  a  week. 

11.  28  inches  to  the  fraction  of  a  yard. 

12.  10  ounces  to  the  decimal  of  a  pound. 

13.  3  quarts  to  the  decimal  of  a  bushel. 


Denominate  Numbers. 


135 


209.  Written  Exercises. 
Add: 

1.    13  lb.    6  oz.  10  oz.  +9  oz.  +  6  oz.  =  25  oz.  =  1  lb.  9  oz. 

5'  lb.     9  oz.      Write  9  ounces  and  carry  1   to   column  of 

25  lb.  10  oz.      pounds.  Ans.  44  lb.  9  oz. 


2.  19  yd.  1  ft. 

2  ft. 
3  yd.  1  ft. 

3.  5  min.  30  sec. 
11  min.  25  sec. 

9  min.  18  sec. 

4.  4  ft.  9  in. 

2  ft.  6  in. 

7  ft.  7  in. 

6.    18  gal.  3  qt. 
9  gal.  1  qt. 

2  qt. 

210.    Subtract: 

1.  81b. 
4  lb.  7  oz. 

2.  15  yd.  1  ft. 

9  yd.  2  ft. 

3.  17  hr. 

9  hr.  50  min. 


A.    1  yd.  1  ft.  1  in. 
2  ft.  9  in. 

5.   25  gal.  1  qt. 
6  gal.  3  qt. 


6. 


7. 


11  bu.  3  pk. 
6  bu.  2  pk. 

2pk. 


Ipk. 
Ipk. 

6qt. 
7qt. 
5qt. 

3  wk. 

6  wk. 
1  wk. 

5  da. 

6  da. 
3  da. 

11  T. 
4T. 

165  lb. 

983  lb. 

1756  lb. 

Change  8  lb.  to  7  lb.  16  oz. 

16  oz.  —  7  oz.  =  9  oz. 
7  lb.  -  4  lb.  =  3  lb. 


Ans.  3  lb.  9  oz. 

6.  89  bu.  2  pk. 
67  bu.  3  pk. 

7.  3  pk.  2  qt. 
2  pk.  7  qt. 

8.  11  wk.  1  da. 

9  wk.  5  da. 

9.  5T.    896  1b. 

1984  lb. 


136  Chapter  Three. 

211.  Multiply: 

1.  12  lb.  7  oz.  x  3 

3  times  7  ounces  are  21  ounces,  or  1  pound  6  ounces.  Write  6 
ounces.  3  times  12  pounds  are  36  pounds,  and  1  pound  to  carry  are 
37  pounds.  Ans.  37  lb.  5  oz. 

2.  3  hr.  10  min.  x  7  7.  7  min.  18  sec.  x  10 

3.  4  T.  985  lb.  x  11  8.  9  gal.  3  qt.  x  2 

4.  7  bu.  3  pk.  x  9  9.  2  ft.  9  in.  x  8 

5.  3  wk.  4  da.  x  4  10.  1  yd.  1  ft.  6  in.  x  6 

6.  4  yd.  1  ft.  x  5  11.  3  yr.  4  mo.  x  7 

212.  Divide: 

1.  9  1b.  2oz.  -s-2 

\  of  9  pounds  is  4  pounds  and  1  pound  remainder,     ^q  •,,     « 

or  16  ounces.     Add  to  this  2  ounces,  giving  18  ounces       ' .      *  _ ■ 

4  lb.  9  oz. 
for  the  dividend.    \  of  18  ounces  is  9  ounces. 

Ans.  4  lb.  9  oz. 

2.  31  gal.  2  qt.  -f-  9  7.    19  ft.  2  in.  -- 10 

3.  19  hr.  21  min.  --3  8.    34  T.  936  lb.  --  4 

4.  26  bu.  1  pk.  4-5  9.    17  wk.  1  da.  4-6 

5.  41  min.  44  sec.  4-  8  10.    52  yd.  0  ft.  9  in.  4- 11 

6.  18  yd.  2  ft.  4- 7  11.   23  yr.  4  mo.  4- 7 

213.  Divide: 

1.   18  lb.  4  oz.  by  4  lb.  9  oz. 

18  lb.  4  oz.  =  292  oz. 
4  lb.  9  oz.  =  73  oz. 
292  oz.  -r-  73  oz.  =  4,  Ans. 

Note.  —  Change  the  divisor  and  the  dividend  to  the  same  denomi- 
nation.   The  answer  is  an  abstract  number. 


Denominate  Numbers.  137 

2.  16  yd.  by  2  yd.  2  ft. 

3.  51  hr.  36  min.  by  6  hr.  27  min. 

4.  47  min.  42  sec.  by  5  min.  18  sea 

5.  84  yr.  7  mo.  by  12  yr.  1  mo. 

6.  19  da.  3  hr.  by  2  da.  3  hr. 

7.  3  mi.  40  rd.  by  125  rd. 

8.  103  T.  808  lb.  by  8  T.  1234  lb. 

9.  52  gal.  2  qt.  by  3  gal.  2  qt. 

10.  68  bu.  1  pk.  by  5  bu.  1  pk. 

11.  30  ft.  8  in.  by  1  ft.  11  in. 

12.  52  yd.  9  in.  by  4  yd.  2  ft.  3  in. 

13.  51  wk.  3  da.  by  2  wk.  6  da. 

214.   Oral  Problems. 

1.  What  will  be  the  weight  of  16  hams  that  average  10 
lb.  5  oz.  each  ? 

2.  From  a  chest  of  tea  containing  54  pounds  there  were 
sold  27  lb.  7  oz.     How  many  pounds  remain  ? 

3.  Seven  bushels  of  potatoes  are  divided  among  8  per- 
sons.    How  many  pecks  and  quarts  does  each  receive  ? 

4.  How  many  square  inches  in  the  surface  of  a  sheet  of 
paper  measuring  11  inches  by  13  inches  ? 

5.  How  many  feet  and  inches  in  £  yard  ? 

6.  What  decimal  of  a  pound  is  14  ounces  ? 

7.  A  man  buys  a  bushel  of  hickory  nuts.     After  he  sells 
2  pk.  4  qt.,  what  fraction  of  the  bushel  has  he  left  ? 

8.  A  dealer  puts   30  gallons   of  milk   in   cans   holding 
1  qt.  1  pt.  each.     How  many  cans  does  he  fill  ? 

9.  At  $  24  per  month,  how  much  rent  will  a  man  pay 
in  1  year  and  5  months  ? 


138  Chapter  Three. 

10.  75  hundredths  of  a  pound  is  how  many  ounces? 

11.  How  many  feet  in  5  rods  ? 

12.  7  qt.  1  pt.  of  milk  is  divided  among  5  people.  How 
many  quarts  and  pints  does  each  receive  ? 

13.  What  fraction  of  2  lb.  3  oz.  is  1  lb.  4  oz.  ? 

14.  Three-eighths  of  a  ton  is  how  many  pounds  ? 

15.  Change  9  hr.  36  min.  to  the  fraction  of  a  day. 

215.   Written  Problems. 

1.  32  hams  weigh  464  pounds.  What  is  the  average 
weight  ? 

2.  595  gallons  of  oil  are  put  into  14  barrels.  How  many 
gallons  and  quarts  does  each  contain  ? 

3.  If  there  are  42  gallons  and  2  quarts  in  a  barrel  of  oil, 
how  much  oil  will  there  be  in  15  barrels  ? 

4.  A  piece  of  cloth  containing  57  yards  is  divided 
equally  among  six  persons.  What  is  the  length  of  each  one's 
share  ? 

5.  How  many  minutes  in  a  day? 

6.  July  1  is  the  last  school  day.  How  many  days' 
vacation  will  there  be,  if  school  begins  September  6  ? 

7.  How  many  hours  and  minutes  are  there  from  half- 
past  3  Saturday  afternoon  to  a  quarter  before  9  Monday 
morning  ? 

8.  How  many  steps,  2  ft.  6  in.  long,  must  a  man  take 
in  walking  1200  feet  ? 

9.  A  man  owns  a  plot  of  ground  420  feet  long,  240  feet 
wide.  How  many  rods  of  fence  will  be  required  to  enclose 
it? 

10.  A  train  goes  from  Jersey  City  to  Washington,  228 
miles,  in  4  hr.  12  min.  How  many  miles  an  hour  does  it 
travel  ?     How  long  does  it  take  the  train  to  go  one  mile  ? 


Measurements.  139 

11.  On  Monday  a  boarding-house  uses  3  gallons  2  quarts 
of  milk ;  on  Tuesday,  4  gallons ;  on  Wednesday,  3  gallons 
1  quart;  on  Thursday,  4  gallons  2  quarts;  on  Friday,  6 
gallons ;  on  Saturday,  5  gallons  2  quarts ;  on  Sunday,  3  gal- 
lons. How  much  is  used  during  the  week,  and  what  is  the 
average  per  day  ? 

12.  June  21  the  sun  rises  at  New  York  at  4.23  a.m.  and 
sets  at  7.40  p.m.     How  long  is  the  night  ? 

13.  From  3£  bushels  take  3  pecks. 

14.  What  is  the  number  of  rods  in  the  perimeter  of  a 
field  206  ft.  3  in.  wide  and  twice  as  long  ? 

MEASUREMENTS. 

216.  Written  Exercises. 

How  many  square  inches  in  each  of  the  following  rec- 
tangles ?     First  change  each  dimension  to  inches. 

1.  42  in.  by  36  in.  6.  9  ft.  by  11  ft. 

2.  71  in.  by  18  in.  7.  27  in.  by  30  in. 

3.  3  ft.  1  in.  by  4  ft.  2  in.  8.  65  in.  by  92  in. 
Note.  —37  in.  by  50  in.  9.  7  ft.  3  in.  by  2  yd. 

4.  5  ft.  3  in.  by  6  ft.  4  in.  10>  3  yd  by  6  ft  6  in 

5.  12  ft.  by  18  ft.  (108  in.  by  78  in.) 

217.  How  many  square  feet  in  each  of  the  following  rec- 
tangles ?  First  change  each  dimension  to  feet,  or  to  feet  and 
a  fraction. 

11.  18  ft.  by  24  ft.  15.   31  ft.  by  4  ft. 

12.  36  in.  by  4  ft.  16-   3  ft.  by  1J  yd. 

(3  ft.  by  4  ft.)  17.    42  in.  by  4  ft. 

13.  6  yd.  by  8  yd.  18.    25  ft.  by  17  ft.  6  in. 

(18  ft.  by  24  ft.)  19.    42  in.  by  48  in. 

14.  1  yd.  by  48  in.  20,    13  yd.  by  15  yd. 


140  Chapter  Three. 

218.  How  many  square  yards  in  each  of  the  following 
rectangles  ?  Change  each  dimension  to  yards,  or  to  yards 
and  a  fraction. 

21.  18  yd.  by  25  yd.  26.  36  yd.  by  24  in. 

22.  15  yd.  by  1  yd.  1  ft.  27.  17  ft.  6  in.  by  32  in. 

23.  27  ft.  by  36  ft.  28.  22  ft.  9  in.  by  18  in. 

24.  54  ft.  by  2  ft.  6  in.  29.  108  in.  by  90  in. 

25.  24  yd.  by  27  in.  30.  180  ft.  by  54  in. 

219.  Oral  Exercises. 

1.  If  a  table  is  3  yards  long  and  2  yards  wide,  how  many 
square  feet  in  it  ? 

2.  If  it  takes  24  yards  of  carpet,  a  yard  wide,  to  cover  a 
floor,  how  many  yards  f  yard  wide  will  be  needed  for  the 
same  floor  ? 

3.  How  many  square  inches  in  \  of  a  square  foot  ? 

4.  A  room  is  21  feet  long  and  18  feet  wide.  What  will  it 
cost,  at  5  cents  per  yard,  for  a  strip  of  moulding  around  the 
walls  ? 

5.  How  many  square  yards  of  carpet  would  be  needed 
for  the  floor  of  the  above  room  ? 

6.  A  field  is  40  rods  long  and  26  rods  wide.  What  is  the 
distance  around  it  ? 

7.  What  will  it  cost  to  carpet  a  room  18  feet  long,  15  feet 
wide,  at  75  cents  per  square  yard  ? 

8.  What  is  the  cost  of  fencing  a  lot  24  rods  long  by  20 
rods  wide,  at  $  1.12 J  per  rod? 

9.  My  field  is  100  rods  long  and  75  rods  wide.  How 
much  is  it  worth  at  $  2  a  square  rod  ?  How  much  will  it 
cost  to  fence  it  at  $  1  a  rod  ? 

10.  How  many  yards  of  fence  will  be  required  to  enclose 
a  rectangular  field  98  yards  long  and  50  yards  wide  ? 


Measurements.  141 

220.  Written  Problems. 

Make  a  diagram  in  each  case  : 

1.  A  lot  25  feet  by  100  feet  has  on  it  a  house  25  feet  by 
55  feet.     How  many  square  feet  are  there  left  for  a  yard  ? 

2.  How  many  square  feet  are  there  in  the  floor  of  a 
room  24  feet  long,  18  feet  wide  ? 

3.  How  many  square  yards  are  there  in  the  ceiling  of 
the  same  room  ? 

4.  Find  the  number  of  square  yards  of  plastering  needed 
for  the  end  wall  of  a  room  18  feet  wide,  9  feet  high,  after 
deducting  for  two  windows  each  6  feet  high,  4£  feet  wide. 

5.  How  many  square  yards  of  plastering  will  be  needed 
for  the  opposite  wall  of  the  same  room,  18  feet  wide,  9  feet 
high,  after  deducting  for  a  door  7J  feet  high,  6  feet  wide  ? 

6.  Calculate  the  number  of  square  yards  of  plastering 
needed  for  two  side  walls  of  a  room  24  feet  long,  9  feet 
high,  after  deducting  for  a  fireplace  6  feet  square  on  one 
side. 

7.  A  house  30  feet  by  60  feet,  with  an  addition  15  feet 
square,  is  built  upon  a  lot  100  feet  square.  How  many 
square  feet  of  ground  are  covered  by  the  building  ?  How 
many  square  feet  remain  for  a  garden  ? 

8.  Measure  the  top  of  a  brick  and  calculate  the  number 
of  square  inches  in  its  surface.  How  many  square  inches 
in  the  surface  of  the  bottom  of  the  brick?  Measure  one 
side,  and  calculate  its  surface.  How  many  square  inches 
are  there  in  the  surface  of  the  opposite  side  ?  How  many 
square  inches  in  each  end  ? 

9.  Measure  a  crayon  box,  and  calculate  the  number  of 
square  inches  in  each  face. 


142 


Chapter  Three. 


10.  Calculate  the  number  of  square  feet  in  the  floor  of 
the  classroom.  In  the  ceiling.  In  each  side  wall.  In  each 
end  wall. 

11.  What  will  it  cost  to  put  moulding  around  a  room 
shaped  like  the  drawing,  allowing  3  inches  on  every  corner 
for  matching,  the  moulding  being  worth  5f  ^  a  foot? 


10  ft. 

• 

£ 

-«*< 

6  ft. 

€ 

05 

22  ft. 

12.  The  circumference  of  a  circle  is  3.1416  times  the 
diameter.  What  is  the  diameter  of  a  circular  track  1760 
yards  in  circumference  ?     Find  to  two  places  of  decimals. 

13.  Show  the  difference  between  2  square  inches  and  2 
inches  square. 

14.  How  many  paving  tiles  6  inches  square  are  needed  to 
cover  a  floor  18  feet  long,  10  feet  wide  ? 

15.  How  many  flagstones,  each  4  feet  long  and  2  feet 
wide,  will  be  needed  to  lay  a  crossing  32  feet  long  and  6 
feet  wide  ?  What  will  be  the  cost  of  them  at  the  rate  of 
$50  for  100  stones? 


Measurements.  143 

AREAS  OF  RIGHT-ANGLED  TRIANGLES. 

221 .   Preliminary  Exercises. 

The  square  shown  in  the  diagram  is  divided  into  two  parts 
by  a  diagonal.     One  side  of  the  square  measures  10  feet. 

1.   Mark  in  each  triangle  its  area. 


Square.  Eectangle. 

2.  Divide  a  rectangle  20  feet  by  12  feet  into  two  parts 
by  a  diagonal.     Mark  in  each  triangle  its  area. 

3.  Draw  a  right-angled  triangle  3  inches  by  4  inches. 
Calculate  its  area  in  square  inches. 

4.  How  many  square  yards  in  the  sur-  /  jj 
face  of  a  right-angled  triangle  whose  base 
measures  30  feet,  and  whose  perpendicular 
measures  224  feet  ?                                               /      n         £ 

222.   Written  Exercises. 

Find  the  area  in  square  feet  of  the  following  right-angled 
triangles.     (Change  each  dimension  to  feet.) 

1 .    Base  20  yards,  perpendicular  30  feet. 

Area  =  1  square  foot  x  £  (60  x  30)  =  900  square  feet,  Ans. 

Tlie  number  of  square  feet  in  the  area  of  a  right-angled 
triangle  is  equal  to  one-half  the  product  of  the  number  of  feet 
in  the  base  by  the  number  of  feet  in  the  perpendicular, 


144 


Chapter  Three. 


2.  Base  16  inches,  perpendicular  3  feet. 

3.  Base  30  inches,  perpendicular  1  yard. 

4.  Base  3  feet  6  inches,  perpendicular  5  feet. 

5.  Base  2  yards  1  foot,  perpendicular  1  yard  9  inches. 

6.  Base  50  yards,  perpendicular  36  yards. 

7.  Base  112J-  feet,  perpendicular  30  yards. 

8.  Base  90  inches,  perpendicular  2  feet. 

9.  Base  12^-  yards,  perpendicular  13^  yards. 

10.  Base  1  rod,  perpendicular  1\  feet. 

11.  Base  33 \  feet,  perpendicular  18  feet  6  inches. 


BILLS. 
223.  Philadelphia,  Sept.  24,  1905. 

Mr.  William  J.  Hurley, 

To  John  J.  Petit  &  Son,  Dr. 


To  50  lb.  Pipe 

5\t 

To  8  Faucets 

75$ 

To  1  Sink 

To  3^  days'  Labor 

94.7S 

75 


$ 


1.  Copy  and  complete  the  above  bill. 

2.  Albert  Janson  has  done  3J  days'  work,  @  $  3.50  per 
day,  for  Ephraim  Whitlock.  He  charges  for  850  feet  lumber, 
at  $  2  per  hundred;  5  pounds  of  nails,  at  9^  per  pound; 
3  locks,  @  50^ ;  2  bolts,  at  10j*.     Make  out  his  bill. 

3.  A  gardener  furnishes  3  rose  bushes,  at  75^;  4  grape- 
vines, at  50^ ;  11  fuchsias,  at  30^ ;  25  pansies,  at  10^.  He 
charges  $3.25  per  day  for  2\  days'  labor.     Make  out  his  bill. 


Percentage.  145 

4„  An  upholsterer  charges  $  3  per  day  for  repairing  some 
furniture.  He  supplies  6  pounds  of  hair,  at  50^  per  pound ; 
17  yards  of  plush,  at  $  1.75  per  yard;  3  papers  of  tacks,  at 
10^;  cord,  gimp,  etc.,  47^.  He  works  4  days.  Make  out 
his  bill. 

Note. — The  foregoing  bills  are  for  work  done  and  materials  sup- 
plied.   Notice  how  the  heading  differs  from  those  in  Arts.  103  and  173. 

5.  Make  out  and  receipt  a  bill  for  four  articles  bought 
to-day  by  John  Harrigan  from  Metz  and  Fagan,  grocers 
(Art.  103). 

6.  Make  out  a  bill  containing  ten  items  bought  by  Mrs. 
A.  S.  Jacobs,  at  different  times  during  October,  1905,  from 
Frederick  Loeser  &  Co.,  dealers  in  dry  goods  (Art.  173). 

7.  Make  out  a  bill  for  labor  done  and  materials  furnished 
by  Joseph  Minew,  gardener. 


PERCENTAGE. 

224.  Per  cent  means  hundredths. 

Six  per  cent  means  six  hundredths,  jfo,  or  .06.    It  is  writ- 
ten 6%. 

225.  Oral  Exercises. 

1.  What  is  6%  of  200? 

6%  means  jfo.    To  find  6%  of  200,  we  multiply  200  by  ^fo,  or 
200  x  .06.  Arts.  12. 

2.  What  is  yffr  of  300  ?  6.  6%  of  150 

3.  Find  .06  of  400  7.  6%  of  250 

4.  6  per  cent  of  500  8.  6%  of  125 

5.  6%  of  50  9.  6<f  of    75 


146  Chapter  Three. 

10.  6%  of   60  16.     \%  of  600 

11.  6%  of  160  17.     \%  of  600 

12.  4%  of  125  18.   2\%  of  600 

13.  7%  of  500  19.   3J%  of  400 

14.  5%  of  240  20.     I  %  of  400 

15.  1%  of  600  21.     9%  of    90 

In  solving  examples  in  percentage,  the  work  is  frequently  shortened 
by  changing  the  per  cent  to  a  common  fraction  in  its  lowest  terms. 

76%  =  *  =  *.  Arts. 

226.  What  fraction  equals  : 

1.  25%  5.   20%  9.     6f% 

2.  121%  6.   50%  10.   37^% 

3.  33$%  7.   6J%  11.   62£% 

4.  16|%  8.   3J%  12.   87£% 

227.  1.   Find  50%  of  96. 

60  %  of  96  =  I  of  96  =  48,  Ans. 

2.  25%  of    72  10.   150%  of  140 

3.  12^%  of  120  11.   250%  of  140 

4.  6J%  of    48  12.    125%  of  140 

5.  33£%  of    36  13.       1%  of  140 

6.  16|%  of  126  14.       1%  of  350 

7.  $±%  of    72  15.       2%  of  350 

8.  100%  of  140  16.   3%  of  350 

9.  200%  of  140  17.   4%  of  350 


Percentage.  147 

228.   Written  Problems. 

Note.  —  The  pupils  should  find  but  little  difficulty  in  solving  these 
problems,  which  will  serve  to  show  a  few  applications  of  percentage. 
There  is  no  need  of  preliminary  explanation  of  terms  the  meaning  of 
which  can  readily  be  determined  from  the  context. 

1.  A  merchant  sells  a  lot  of  cotton  for  f  1872.50.  He 
receives  2%  of  this  amount  for  selling  it.  How  much  does 
he  receive  ?  He  receives  $  1872.50  x  .02. 

2.  How  much  will  it  cost  me  to  insure  goods  to  the 
amount  of  $  18,760  at  one  per  cent  ? 

3..  A  dealer  imports  books  worth  $  548.40,  on  which  he 
pays  duty  to  the  government  at  the  rate  of  25%.  What  is 
the  amount  of  the  duty  ? 

4.  Eighty  per  cent  of  a  class  of  55  pupils  are  promoted. 
How  many  are  not  promoted  ? 

5.  A  man  buys  a  house  for  $  16,000  and  sells  it  at  an 
advance  of  3  per  cent  over  the  cost.  How  much  does  he 
gain  ? 

6.  A  clerk  spends  for  rent  18  per  cent  of  his  income  of 
$  1850  per  year.     What  rent  does  he  pay  ? 

7.  A  girl  spelled  correctly  95  per  cent  of  60  words. 
How  many  did  she  miss? 

8.  Tea  costing  40  cents  per  pound  is  sold  at  a  profit  of 
50  per  cent.     What  is  the  selling  price  ? 

9.  I  loan  $  600  at  6%  interest  per  year.  How  much  in- 
terest should  I  receive  from  January  1,  1903,  to  January  1? 
1905? 

10.  I  loan  a  person  $  600  on  July  1,  1903.  He  agrees  to 
pay  me  5%  of  the  amount  loaned  per  year  as  interest.  How 
much  interest  should  I  receive  July  1,  1904  ? 

11.  A  house  is  valued  at  $6000.  How  much  taxes  must 
the  owner  pay  at  the  rate  of  $1.25  per  $100  valuation  ? 


148  Chapter  Three, 

INTEREST. 
229.   Oral  Exercises. 

Note.  —  A  preliminary  talk  with  the  class  should  develop  the  fact 
that  a  person  hiring  a  horse  is  charged  for  its  use,  say  so  much  an 
hour ;  that  a  person  hiring  a  house  is  charged  so  much  a  month  or  a 
year  for  its  use.  A  person  borrowing  money  is  also  charged  for  the 
use  of  money.  As  the  sum  charged  for  the  rent  depends  upon  the 
size  and  value  of  the  house,  so  the  sum  charged  for  the  use  of  money 
depends  upon  the  sum  loaned. 

A  charge  for  use  of  money  is  called  interest.  The  sum  on 
which  the  interest  is  paid  is  called  the  principal.  The  price 
or  rate  is  a  certain  per  cent  for  a  year. 

1.  What  will  be  the  interest  on  $  100  for  1  year  at  4%  ? 

$  100  x  .04  =  $  4,  Ans. 

2.  On  9  200  for  a  year  at  5%  ? 

3.  On  1 300  for  a  year  at  6%  ? 

4.  On  $  400  for  a  year  at  7%  ? 

5.  On  $  250  for  a  year  at  4%  ? 

At  4%  per  year,  what  will  be  the  interest : 

6.  On  $  200  for  1  year? 

7.  On  $  300  for  2  years  ? 

8.  On  $  100  for  3  years  ? 

9.  On  f  200  for  1\  years  ? 

10.  On  $200  for  1  year  6  months  ? 

11.  What  will  be  the  interest  on  $  200  for  3  years  at  5%  ? 
The  interest  for  1  year  will  be  $  200  x  .05,  or  $  10  ;  for  three  years 

it  will  be  3  times  $  10,  or  $  30,  Ans. 

12.  On  $  300  for  2  years  at  6%  ? 

13.  On  $  400  for  6  years  at  3%  ? 

14.  On  $  100  for  5  years  at  7%  ? 


Interest.  149 

15.  On  $  250  for  2  years  at  4%  ? 

16.  On  $  100  for  1  year  6  months  at  6%  ? 

17.  On  $  200  for  3  months  at  4%  ? 

At  4%  per  year,  what  will  be  the  interest: 

18.  On  $  200  for  6  months  ? 

$200x.04xi. 

19.  On  1 300  for  4  months  ? 

20.  On  |  400  for  3  months  ? 

21.  On  $  300  for  2  months  ? 

22.  On  9  150  for  1  month  ? 

23.  Find  the  interest  on  $  24  for  1  year  at  5%. 

24.  On  $  36  for  1  year  at  4%. 

25.  On  $  67  for  1  year  at  3%. 

230.   "Written  Exercises. 
Find  the  yearly  interest  on : 

1.  $286.50  at  4%  $286.50 

Multiply  the  principal  by  the  rate,  4  %,  written  as       _ 

a  decimal.  $  11.4600 

Ans.   $11.46. 

2.  $  485  at  6%  12.  $  168  at  3|  % 

3.  $375.40  at  5%  13.  $244at5j% 

4.  $  379  at  3%  14.  $  890  at  7T\% 

5.  $  486  at  4i%  15.  $  63.75  at  4% 

6.  $186.75  at  4%  16.  $937.50  at  6% 

7.  $199.50  at  2%  17.  $  980.40  at  5% 

8.  $636  at  3J%  18.  $159.60  at  2\% 

9.  $84.70  at  6%  19.  $  1357.37  at  7% 

10.  $  93.25  at  8%  20.    $  2146.18  at  H% 

11.  $1257  at  7%  21.    $369.40  at  3|% 


150  Chapter  Three. 

Find  the  interest  on  : 

$290  for  2  years  at  4%. 

The    interest    for    1    year    is      $  290. 
$11.60.      Multiplying   by    2,    we  .04 

get  the    interest    for    2    years,        $  11.60  interest  for  1  year 

$23.20.  2 

Ans.  $  23.20  interest  for  2  years 

Multiply  the  principal  by  the  rate  expressed  as  hundredths, 
and  this  product  by  the  time  expressed  in  years  and  fraction 
of  a  year. 

22.  $  1400  for  3  years  at  4|%. 

23.  $  2840  for  4  years  at  5%. 

24.  $  1250  at  6%  for  3  years. 

25.  $  5360  at  5-|%  for  2  years. 

26.  $  380  at  3%  for  4J  years. 

27.  $  780  for  1  year  4  months  at  6%. 
Note.  —  1  year  4  months  =  1£  year. 

28.  $  2560  for  2  years  6  months  at  5%. 

29.  $  1025  for  3  years  3  months  at  4%. 

30.  $  1296  for  7  months  at  7%. 
Note.  —  7  months  =  ^  year. 

31.  $  648  for  5  months  at  5%. 

32.  $  275  for  4  months  at  3%. 

33.  $  1000  for  11  months  at  6%. 

231.   Oral  Problems. 

1.  I  bought  a  house  for  $4000,  and  sold  it  for  80%  oi 
the  cost.     For  what  did  I  sell  it  ? 

2.  A  merchant  whose  income  is  $2000  a  year  spends 
75%  of  it.     How  much  does  he  save? 


Review.  151 

3.  John  has  $  30  in  the  bank,  Mary  has  16f  %  as  much. 
How  much  has  Mary  ? 

4.  If  I  buy  goods  for  $  400  and  sell  them  at  a  loss  of 
5%,  how  much  do  I  lose  ? 

5.  A  farmer  had  100  sheep  and  sold  20%  of  them.  How 
many  did  he  sell  ? 

6.  Cloth  shrinks  5%  of  its  length  in  sponging.  What 
is  the  shrinkage  of  a  piece  which  contained  40  yards  before 
sponging? 

7.  In  a  school  of  400,  60%  are  boys.  How  many  girls 
in  the  school  ? 

8.  What  is  the  interest  on  $  100  for  2  years  at  4%  ? 

9.  What  is  the  interest  on  $  50  for  one  year  at  6%  ? 
10.   What  is  the  interest  on  $  200  for  2  years  at  3J%  ? 

232.   Written  Problems. 

1.  A  man  receives  a  salary  of  $1800  a  year;  he  pays 
15%  of  it  for  board,  8J%  for  clothing,  and  16%  for  other 
expenses.     What  are  his  yearly  expenses  ? 

2.  My  expenses  during  the  month  of  April  were  $  185.68 ; 
my  expenses  in  May  were  12-|%  less  than  in  April.  What 
were  my  expenses  in  May  ? 

3.  A  lawyer  collected  80%  of  a  debt  of  $2360  and 
charged  5%  commission  on  the  sum  collected.  How  much 
did  the  creditor  receive  ? 

4.  A  house  was  insured  for  $3600  at  1|%.  What  was 
the  cost  of  the  insurance  ? 

5.  What  is  the  interest  on  $  550  for  2  years  6  months 
at  4%  ? 

6.  What  is  the  interest  on  $  1200  for  3  years  at  5%  ? 

7.  A  merchant  sold  goods  that  cost  $  2180  at  a  gain  of 
33£%.     How  much  did  he  receive  for  them  ? 


152  Chapter  Three. 

8.  What  is  the  interest  on  $  720  for  1  year  6  months 
at7%? 

9.  I  bought  1260  pounds  of  sugar  at  4 J-  cents  a  pound 
and  sold  it  at  a  gain  of  10%.     How  much  did  I  sell  it  for  ? 

10.   What  is  the  interest  on  $  350  for  2  years  at  3J%  ? 

APPROXIMATIONS. 

These  approximation  examples  should  not  be  neglected.  Pupils, 
besides  finding  them  useful  in  preventing  gross  errors  in  their  calcula- 
tions, will  be  enabled  later  to  obtain  exact  results  to  similar  examples 
by  an  extension  of  the  methods  used  in  obtaining  approximate  results. 
In  a  following  chapter  will  be  found  suggestions  as  to  the  product  by 
99,  24,  99|,  etc. 

Some  pupils  can  probably  give  the  exact  answer  to  No.  5  —  96  lb. 
at  25f  would  be  $  24;  at  \f  less  per  lb.,  the  cost  would  be  VL?  (\f  x  96) 
less  than  $  24.  The  exact  answers  to  Nos.  2,  3,  8,  9,  and  10  can  be 
obtained  in  a  similar  manner. 

After  the  examples  have  been  used  for  sight  exercises  in  approxi- 
mate answers,  they  should  be  solved  for  the  exact  answers. 

Suggestions.  —  (1)  24  @  |f  (2)  24  @  $125.  (3)  64  @  $}. 
(4)485@$1.     (11)  $27-4-$±.      (12)  $800 +  11  J.      (13)  $24-*-$$. 

233.   Give  approximate  answers,  at  sight: 

1.  23|  lb.  of  tea  @  50^. 

2.  24  horses  @  $  124.95. 

3.  64  yd.  of  carpet  @  87^0. 

4.  485  bu.  of  wheat  @  99f  £ 

5.  96  lb.  of  coffee  @  24^. 

6.  840  yd.  of  dress  goods  @  33^. 

7.  360  yd.  of  oil  cloth  @  66f£ 

8.  48  cwt.  of  straw  @  62f£ 

9.  92  hats  @  $1.49f. 

10.   128  lb.  of  lard  @  12f£ 


Review.  1 53 

234.  Give  approximate  answers  in  whole  numbers : 

11.  $27-5-24^  21.   17.3x3.98765 

12.  $  299.96  -T-  $  1.49 J  22.   256.008  x  .249875 

13.  $  24.05  -*-  37^  23.    25.1234  x  15.93 

14.  $  15.03  -r- 12|^  24.    6.12  x  6.12 

15.  $  60  -v-  $  2.49|f  25.    86.4 x. 996 

16.  $  32  -4-  33-f^  26.    33.333  x  5.004 

17.  $  69.95  ^  87^  27.   799.387  x  .125 

18.  9  60  -T-  62^  28.   7.999  x  7.99 

19.  $  64  -*-  66^  29.   7.33  x  11.0083 

20.  $  27.95  -4-  $  1.75  30.   64.002  x  .3750 

SPECIAL  DRILLS. 

Note.  —  It  is  important  for  pupils  to  keep  up  their  previous  practice 
in  handling  large  numbers  without  a  pencil,  and  to  increase  the  size  of 
the  numbers  from  year  to  year. 

To  add  135  and  89,  the  pupil  first  adds  80,  then  9. 

135  +  80(215)+9  =  224 

235.  Give  sums : 

256  +  56  576  +  76  437  +  73  832  +  99 

394  +  77  646  +  85  768  +  48  543  +  78 

690  +  450  =  690  +  400  +  50 

350  +  680  440  +  590  570  +  640  750  +  250 

770  +  260  '620  +  480  330  +  880  980  +  670 


154  Chapter  Three. 

236.  Give  differences : 

To  subtract  56  from  312  ;  first  deduct  50,  then  6. 

312  -  56  =  312  -  50  (262)  -  6  =  256 

224-89      652-76      500-73      931-99 

471-77      731-85      816-48      621-78 

1200  -  610  =  1200  -  600  -  10 

1140  -  690      1130  -  870      1210  -  570 

1030-350      1100-620      1650-980 

237.  Give  products : 

98  x  4  =  90  x  4  (360)  +  8x4  (32)  =  392 
89  x  5  67  x  7  98  x  4  79  x  3 

78  x  6  75  x  9  66  x  8  89  x  2 

238.  Oral  Problems. 

Note.  —  These  problems  should  first  be  solved  as  sight  exercises 
from  the  book.  Afterward,  one  should  be  read  by  the  teacher  and  the 
answer  written  by  all  the  pupils  at  a  given  signal.  These  problems 
require  no  analysis.  They  contain  numbers  similar  to  those  of  the 
special  drills  on  the  previous  page. 

1.  I  sold  375  bushels  of  wheat  to  one  miller  and  87  to 
another.     How  many  bushels  did  I  sell  ? 

2.  Bought  goods  to  the  amount  of  $4.29.  How  much 
change  from  a  $  5  bill  ? 

3.  What  will  be  the  cost  of  89  tons  of  coal  at  $  5  per 
ton? 

4.  If  49  hats  cost  $  147,  what  is  the  cost  of  one  hat  ? 

5.  567  marbles  are  divided  among  9  boys.  How  many 
does  each  receive  ? 

6.  What  will  be  the  cost  of  a  barrel  of  flour  at  $5.25 
and  8  pounds  of  sugar  at  6^  ? 


Review.  155 

7.  How  much  must  be  paid  for  55  pounds  of  raisins,  at 
8^  per  pound  ? 

8.  Find  the  cost  of  320  pounds  of  hay  at  60^  per  hun- 
dred pounds. 

9.  A    father   earned    $14.60,   his    son    earned    $7.80. 
What  were  the  earnings  of  both  ? 

10.  There  are  36  inches  in  a  yard.  How  many  yards 
are  there  in  324  inches  ? 

11.  The  product  is  925,  the  multiplier  is  25.  What  is 
the  multiplicand  ? 

12.  What  price  was  paid  for  20  sheep,  at  $8.75  per 
head? 

13.  A  man  saved  $320  per  year  for  5  years.  How 
much  more  would  he  require  to  make  $  2000  ? 

14.  Mr.  Jones  sold  a  lot  for  $675,  thereby  losing  $85. 
What  did  he  pay  for  it  ? 

RATIO. 

239.   Written  Problems. 

Note.  —  Indicate  operations,  and  cancel  where  possible. 

1.  If  56  men  can  pave  a  street  in  24  days,  how  long  will 
it  take  32  men  to  pave  it  ? 

Analysis.  — One  man  will  take  56  times  as  long  as  56  men  ;  and  32 
men  will  do  the  work  in  ff  of  the  time  required  by  56  men. 

Problems  of  this  kind,  involving  only  multiplication  and  division,  are 
sometimes  shortened  by  cancellation.  Instead  of  multiplying  24  days 
by  66,  and  dividing  the  product  by  32,  the  pupil  should  indicate  these 

6  7 

operations,  then  cancel :  ?*  days  x  ^  =  42  days,  Arts. 

i 


156  Chapter  Three. 

2.  When  a  vessel  sails  168  miles  a  day,  she  completes 
her  voyage  in  14  days.  In  what  time  would  she  complete 
it  if  she  sailed  196  miles  a  day  ? 

At  168  miles  per  day,  the  voyage  requires  14  days. 
At  196  miles  per  day,  it  would  require  14  days  x  |f§. 

3.  If  a  field  would  support  64  sheep  for  21  days,  how 
long  would  it  support  48  sheep  ? 

4.  If  42  men  could  build  a  wall  in  24  days,  how  many 
men  could  build  it  in  18  days? 

The  pupil  must  first  determine  what  is  asked.  In  this  problem,  it 
is  the  number  of  men.  The  given  number  of  men,  42,  must  first  be 
written  in  the  multiplicand. 

To  build  a  wall  in  24  days  requires  42  men.  To  build  it  in  a  shorter 
time  would  require  more  men,  hence  the  ratio  is  ff . 

5 .  If  21  horses  are  worth  as  much  as  35  cows,  how  many 
horses  are  worth  as  much  as  55  cows  ? 

6.  A  girl  that  wrote  36  letters  to  a  line,  took  15  lines  in 
writing  a  piece  of  dictation.  How  many  lines  would  a  girl 
that  wrote  30  letters  to  a  line  require  for  the  same  dictation  ? 

7.  If  a  boy  that  steps  27  inches  at  a  time  takes  1000  steps 
in  going  home  from  school,  how  many  steps  will  be  taken 
by  a  boy  that  steps  30  inches  ? 

8.  If  1920  bricks  will  build  a  wall  15  yards  long,  how 
many  bricks  will  be  required  for  a  similar  wall  24  yards 
long? 

9.  A  train  going  44  miles  an  hour,  went  a  certain  distance 
in  9  hours.  How  long  would  it  take  a  train  going  36  miles 
an  hour  to  make  the  same  trip  ? 

10.    Find  the   cost   of  one-fourth   of  a  barrel   of  flour 
at  the  rate  of  22  cents  for  7  pounds. 
A  barrel  of  flour  weighs  196  pounds. 


Review.  1 57 

11.  Six  men  can  do  a  certain  piece  of  work  in  eighteen 
days.  How  long  would  it  take  eighteen  boys  to  do  the 
same  work,  if  one  man  can  do  as  much  work  as  two  boys  ? 

12.  If  a  certain  quantity  of  flour  will  last  48  persons  57 
days,  how  .long  will  it  last  38  persons  ? 

SHORT  METHODS. 

240.  Sight  Exercises. 

1.    68x25  68  x  25  =  \  of  6800 

9*  v  4.Q  25  x  49  =  49  x  25  =  J  of  49  hundred 

Z.    ZOX^y  =  12i  hundred  =  1225,  ^na. 

3.  88  X  12 J  i  of  88  hundred 

4.  24x75  13.  48x37£ 

5.  82xl2J  14.  92x50 

6.  72x25  15.  32x33J 

7.  25x51  16.  88x25 

8.  66x33£  17.  25x97 

9.  48  x  75  18.  16  x  87J 

10.  24  x  62£  19.   66  x  66% 

11.  96x25  20.   16x66£ 

12.  25x81  21.   18xl6| 

241.  Written  Exercises. 

1.  9347  x25  934700-4 

2.  863x75  (86300  x  3)  ~  4 

3.  8123  X  12£  812300-8 

Dividing  8123  hundred  by  8  gives  a  quotient  of  1015f  hundred,  the 
fraction  of  which  the  pupil  should  write  at  once  as  37^  units  without 
dividing  out  300  units  by  8.  While  he  sets  down  the  work  in  this  way, 
8)812300,  he  should  be  able  to  write  the  remainder  of  the  answer  when 

1015 
he  reaches  the  annexed  ciphers. 


158  Chapter  Three. 

4.  6483x33  J  |  of  6483  hundred 

5.  8123  x  125  i  of  8123000 

6.  9347x250  14.       33J  x  3870 

7.  9347  x2£*  15.       66|x3456 

8.  9347x75  16.       16f  x  1266 

9.  6483  x66f  17.   8408    x62£ 

10.  6488  x37£  18.   3875    x  37 J 

11.  4896  x87|  19.   1925    x  12£ 

12.  1284  x  62J  20.    7314    x  250 

13.  75x2468  21.   6480    x  125 

242.     Oral  Problems. 

1.  What  will  be  the  cost  of  49  pounds  of  coffee  at  25^ 
per  pound  ? 

2.  I  paid  $14.75  for  eggs  at  25^  a  dozen.     How  many 
dozen  did  I  buy  ? 

3.  What  will  be  paid  for  88  bushels  of  wheat  at  87^ 
per  bushel  ? 

4.  How  many  bushels  of  corn  at  621^  per  bushel  can  be 
bought  for  $  150  ?  ($  i50  +  1 $) 

5.  How  much  will  be  paid  for  99  yards  of  dress  goods 
at  33 £ f  per  yard? 

6.  How  many  yards   of  carpet  at  66%jt  per  yard  can 
be  bought  for  $84? 

7.  Find  the  cost  of  15  dozen  collars  at  12 J^  each. 

8.  Paid  $24  for  cuffs  at  16|^  per  pair.     How  many 
dozen  pairs  were  bought  ? 

9.  What  will  be  the  cost  of  128  pounds  of  tea  at  75^ 
per  pound  ? 


Review.  1 59 


10.  A  bale  of  cotton  at  6 \y  per  pound  cost  $  25.  What 
was  the  weight  of  the  cotton  ? 

11.  A  farmer  sold  hay  at  75^  per  hundredweight,  receiv- 
ing for  it  $  39.     How  many  hundredweights  did  he  sell  ? 

12.  How  many  barrels  of  mess  pork  at  $12.50  per  barrel 
can  be  bought  for  $175? 

13.  What  will  be  the  cost  of  84  yards  of  carpet  at  $  1.25 
per  yard  ? 

14.  When  wheat  sells  at  $  1.12 \  per  bushel,  how  many 
bushels  can  be  bought  for  $  199  ? 

15.  At  $3.50  each,  what  will  be  paid  for  42  coats  ? 

16.  Find  the  cost  of  28  hats  at  $2.75  each. 

17.  A  real  estate  agent  sold  97  lots  at  $250  each.  How 
much  did  he  receive  for  them  ? 

($250  =  \  of  $1000) 

18.  What  will  be  the  cost  of  248  horses  at  $125  each  ? 

19.  At  ^  cent  each,  how  many  penholders  can  I  buy  for 
$4.32? 

20.  Paid  $3075  for  cows  at  $75  each.  How  many  were 
bought  ? 

REVIEW  OF  FRACTIONS. 

Note.  —  Practice  in  the  sight  work  such  as  is  given  in  the  following 
examples  will  enable  pupils  to  dispense  with  some  of  the  aids  they 
found  necessary  to  employ  during  the  earlier  stages  of  work  in  frac- 
tions. These  exercises  should  be  answered  one  at  a  time  from  the 
book  or  the  blackboard,  preferably  the  latter.  At  a  later  lesson,  the 
teacher  should  require  the  answers  to  five  or  ten  examples  selected  pro- 
miscuously, to  be  written  from  the  book  or  the  blackboard,  the  ex- 
amples to  be  announced  by  the  teacher  by  number.  At  the  same,  or 
another  lesson,  the  teacher  should  read  a  few,  the  answers  to  be 
written  one  at  a  time.  In  these  examples  pupils  should  not  take  pen 
or  pencil  until  the  signal  is  given  to  write  the  answer.  No  change 
should  be  made  in  an  answer  after  it  is  written. 


160  Chapter  Three. 

243.  Write  answers  at  sight : 

1.  Add  32£  and  15f . 

Mentally  changing  the  fractions  to  twelfths,  the  pupil  proceeds  as 
follows  :  ^  +  15(47^)  +  &  =  47ft  =  48^,  Ans. 

2.  24£  +  15f.  4.   62{  +  23^. 

3.  50f +  20f  5.   40f +  33f 

6.  From  78-J  take  20£. 

Suggestion.  —  20|  from  78  (and  £)  leaves  57 \  (and  |),  57£  +  |. 

Ans.  67^. 

7.  80£-40£.  9.   33£-16J. 

8.  43|-12£.  10.   54|-30£. 

11.  Multiply  20|  by  6. 

Six  times  20(120)  +  6  times  f  (4)  =  124,  Ans. 

12.  12fx8.  14.    12|x9. 

13.  30f  x  10.  15.    11J  X  6. 

16.  Divide  24f  by  2. 

i  of  24(12) +io£|a)  =  12i,  Jns. 

17.  48^  +  6.  19.  80|-l-4. 

18.  23^-^-3.  20.  55f£^5. 

21.  Divide  24£  by  4. 

iof24(6)  +  Jof|(^)  =  6A,  ^ 

22.  60^3.  24.  28f  +  7. 

23.  40^-^4.  25.  36£  +  9. 

26.  Divide  21J  by  5. 

21$  contains  5,  4  times,  with  a  remainder  of  1J,  or  5  fourths. 
6  fourths  -=-5=1  fourth.  Ans.  4 J. 

27.  17^  +  4.  29.   26J-5-8. 

28.  19£-i-6.  30.    19£-*-3. 


Review.  161 

31.  Divide  18f  by  7. 

18$  -*-  7  =  2,  with  4f  remainder.  |  of  4$  =  \  of  ^  =  U-    An8-  2H« 

32.  25J-5-2.  34.    191-5-4. 

33.  31^ -v- 3.  35.    22J-J-5. 

244.  Written  Exercises. 
Perform  indicated  operations : 

i.  (lj  +  D  +  cef  +  i)  5.  52£X  (ij-W) 

i  +  *  '  5 

'   fof4^|oflf  '15-8 

4.   231-^(3^  +  11)  8.   ^of(3f-2|  +  9i) 

245.  Find  answers : 

9.    Simplify  l±^zM. 

10.  Find  the  sum  of  f,  ^,  £,  &  ^. 

11.  Keduce     ~T  +  i~?  to  a  simple  fraction. 

12.  Divide  (Hi  +  !)by(fx}ix  W- 


13.    Simplify  i±i  of  }  of  ] 


71  —  31 

14.  iz 2s  =  9 

15.  Find  the  value  of  4fr  +  (t  of  tV)  . 

(|ofl»-f 

16.  (2J  +  l|)  +  (2i  +  3i)  =  ? 

17.  Find  the  value  of  2\  times  the  quotient  of  (3  —  2J)- 

18.  3f  +  14-7f  +  5— ft-? 


r6i  Chapter  Three. 

246.   Multiply.     Do  not  reduce  to  improper  fractions. 
12f 


X4i 

51 

4  times  12f  =  51 ;  £  of  12f  =  4*. 

41 

m\ 

i. 

18fx6J                               6.    16fx7£ 

2. 

25f  x8J                             7.   48fxl2£ 

3. 

16fx5£                              8.   37fxl0£ 

4. 

36£x9£                              9.   36fx9J 

5. 

221x6^                          10.   32|x8| 

247.   Oral  Eeview  Problems. 

1.  What  per  cent  does  a  boy  receive  if  he  solves  16 
examples  of  the  20  given  out  ? 

2.  What  is  the  interest  on  $  200  at  4%  for  2  years  ? 

3.  If  2|  yards  of  calico  cost  22  cents,  how  many  yards 
can  be  bought  for  60  ^  ? 

4.  How  old,  Dec.  1, 1904,  was  a  boy  born  Sept.  1, 1891  ? 

5.  What  is  the  cost  of  3500  bricks  at  $  6  per  M  ? 

6.  How  many  sheep,  at  $5  each,  should  be  given  in 
exchange  for  12  horses,  worth  $  200  each  ? 

7.  75  men  can  do  a  certain  piece  of  work  in  9  days. 
How  long  will  it  take  45  men  to  do  the  same  work  ? 

8.  If  4  barrels  of  oil  each  containing  42  gallons  are 
emptied  into  a  tank  of  200  gallons'  capacity,  how  many 
more  gallons  will  the  tank  hold  ? 

9.  Change  .375  yard  to  feet  and  inches. 

10.  How  many  half-pints  in  2  gal.  1  qt.? 

11.  How  many  eggs  in  15  dozen  and  6  eggs  ? 


Review.  i6j 

12.  f  =  how  many  98ths  ? 

13.  Find  the  greatest  common  divisor  of  12,  18,  27. 

14.  Find  the  least  common  multiple  of  8,  9,  12. 

15.  How  many  yards  in  5  pieces  of  cloth,  each  contain- 
ing 12J  yards  ? 

16.  Divide  29|  by  7. 

17.  "When  silk  is  75^  per  yard,  how  many  yards  can  be 
bought  for  $9.75? 

18.  If  2|  yards  ribbon  cost  42  cents,  what  will  3f  yards 
cost? 

19.  If  eggs  are  sold  at  the  rate  of  18  for  25  cents,  what 
will  be  the  cost  of  6  dozen  eggs  ? 

20.  Three  men  require  22  days  to  do  a  certain  piece  of 
work.  How  long  would  it  take  11  men  to  do  the  same 
work? 

21.  A  farmer  divides  his  farm  of  425  acres  into  fields  of 
12  J  acres  each.     How  many  fields  has  he  ? 

22.  What  will  be  the  cost  of  46  tons  of  hay,  at  $  12 J  per 
ton? 

23.  What  is  the  weight  of  25  firkins  of  butter,  each  con- 
taining 56  pounds  ? 

24.  At  $  1.75  per  yard,  how  many  yards  of  cloth  can  be 
bought  for  $49? 

25.  If  the  interest  of  $  1  is  6^  a  year,  what  is  the  interest 
of  three  dollars  for  two  years  ? 

26.  If  4  boxes  of  raisins  cost  $7,  what  will  12  boxes  cost? 

27.  A  man  having  75  dollars  bought  7  sheep,  and  had  $5 
left.     What  did  he  pay  for  each  sheep  ? 

28.  A  boy  had  59  peaches  and  found  22  more ;  he  then 
divided  all  of  them  equally  among  9  boys.  How  many  did 
he  give  to  each  ? 


164  Chapter  Three. 

29.  I  bought  2  J  pounds  of  sugar  at  one  store  and  3  J 
pounds  at  another.    How  many  pounds  did  I  buy  in  all  ? 

30.  If  I  of  a  load  of  hay  is  worth  $  14,  what  will  two 
loads  be  worth  ? 

31.  2J  +  1J  =  ?  32.   2fxl£=? 

33.  f  of  my  money  equals  63^.     What  is  %  of  it  ? 

34.  Least  common  multiple  of  8, 12,  15,  24? 

35.  If  5  men  can  do  a  piece  of  work  in  12  days,  in  how 
many  days  can  3  men  do  twice  as  much  work  ? 

36.  John  lost  \  of  his  money  and  has  96^  left.  How 
much  had  he  at  first  ? 

37.  At  6^  a  quart,  what  will  10  quarts  1  pint  of  milk  cost  ? 

38.  I  bought  a  dozen  oranges  at  the  rate  of  4  oranges  for 
3$,  and  sold  them  at  the  rate  of  3  oranges  for  0.  How 
much  did  I  make  ? 

39.  How  long  would  it  take  3  men  to  cut  12  cords  of 
wood,  if  4  men  can  cut  8  cords  in  2  days  ? 

40.  John  sold  24  tops  at  the  rate  of  3  tops  for  ten  cents, 
and  with  the  money  bought  pictures  at  8^  each.  How  many 
pictures  did  he  buy  ? 

41.  How  many  pounds  of  cheese  at  -fa  of  a  dollar  per 
pound  can  be  bought  for  j  of  a  dollar  ? 

42.  18  is  f  of  -£  of  what  number  ? 

43.  If  one  man  can  do  a  piece  of  work  in  llf  days,  in 
what  time  can  12  men  do  it  ? 

44.  How  many  times  is  £  contained  in  2 J  ? 

45.  If  oranges  are  37  J  cents  per  dozen,  what  will  be  the 
cost  of  a  box  containing  480  oranges  ? 


Review.  165 

248.   Written  Eeview  Problems. 

1.  At  70  cents  per  100  pounds,  what  will  be  the  amount 
of  duty  on  an  invoice  of  3622  steel  rails,  each  rail  being 
27  feet  long  and  weighing  60  pounds  to  the  yard  ? 

2.  A  man  had  property  valued  at  $6500.  What  will 
be  his  taxes  at  the  rate  of  $10.80  per  $1000  ? 

3.  Multiply  seventy  thousand  fourteen  hundred-thou- 
sandths by  one  hundred  nine  millionths,  and  divide  the 
product  by  five  hundred  forty-five. 

4.  What  number  multiplied  by  43f  will  produce  265f  ? 

5.  What  decimal  of  a  bushel  is  3  quarts  ? 

6.  A  man  sells  f  of  an  acre  of  land  for  $  93.75.  What 
would  be  the  value  of  his  farm  of  150f  acres  at  the  same 
rate? 

7.  A  coal  dealer  buys  375  tons  coal  at  $4.25  per  ton  of 
2240  pounds.  He  sells  it  at  $  4.50  per  ton  of  2000  pounds. 
What  is  his  profit  ? 

8.  Bought  60  yards  of  cloth  at  the  rate  of  2  yards  for  $  5, 
and  80  yards  more  at  the  rate  of  4  yards  for  $  9.  I  imme- 
diately sold  the  whole  of  it  at  the  rate  of  5  yards  for  $  12. 
How  much  did  I  gain  ? 

9.  A  man  purchased  40  bushels  of  apples  at  $1.50  per 
bushel.  Twenty-five  hundredths  of  them  were  damaged, 
and  he  sold  them  at  20  cents  per  peck.  He  sold  the 
remainder  at  50  cents  per  peck.  How  much  did  he  gain 
or  lose  ? 

10.  If  oranges  are  37^-  cents  per  dozen,  how  many  boxes, 
each  containing  480,  can  be  bought  for  $  60  ? 

11.  A  man  can  do  a  piece  of  work  in  18f  days.     What 
part  of  it  can  he  do  in  6|  days  ? 

12.  How  old  to-day  is  a  boy  that  was  born  Oct.  29, 1896  ? 


1 66  Chapter  Three. 

13.  At  the  rate  of  $  5  per  ton,  what  should  be  paid  for 
125  pounds  of  coal  ? 

14.  From  ten  and  five  hundredths  take  the  sum  of  six 
ten-thousandths  and  15  millionths,  multiply  the  remainder 
by  one-tenth,  and  divide  the  product  by  5000. 

15.  Keduce  the  following  common  fractions  to  decimals, 
and  perform  the  operations  indicated : 

("oWC  X  ■  "2U")  ~-~  2  00000* 

16.  A  man  died  in  1903,  aged  94;  his  son  died  in  1887, 
aged  47.     How  old  was  the  man  at  the  birth  of  his  son  ? 

17.  Multiply  the  sum  of  6§  and  4|  by  their  difference. 

18.  What  will  be  the  cost  of  86,400  feet  of  gas  at  $1.25 
per  thousand  feet  ? 

19.  What  time  elapsed  between  the  discovery  of  America, 
Oct.  14,  1492,  and  Jan.  1,  1904  ? 

20.  How  many  hats  can  be  bought  for  $237.25,  at  the 
rate  of  $13  per  dozen  ? 

21.  A  clerk  receives  a  salary  of  $1500  per  year,  and  his 
expenses  are  $968.  In  what  time  can  he  save  enough  to  buy 
133  acres  of  land  at  $28  per  acre  ? 

22.  What  will  be  the  rent  of  a  house  for  1  yr.  10  mo.  at 
$45  per  month? 

23.  The  product  is  .00087,  the  multiplicand  is  7.25.  What 
is  the  multiplier  ? 

24.  A  man  sells  cloth  at  $2.88  per  yard,  losing  .04  of  the 
cost.     How  much  did  he  pay  per  yard  ? 

25.  A  farm  hand  agreed  to  work  for  $300  per  year  and  a 
horse  worth  $60.  If  he  leaves  at  the  end  of  9  months,  how 
much  is  due  him  if  he  has  already  received  $100  and  the 
horse  ? 


Review.  167 

26.  A  train  running  36  miles  per  hour  leaves  a  station  at 
9  a.m.  At  10.30  a.m.  a  second  train  leaves  and  runs  at  the 
rate  of  30  miles  per  hour.  How  many  miles  apart  are  the 
trains  at  noon,  if  they  run  in  the  same  direction  ? 

27.  Multiply  twenty  thousand  nine  hundred  eight  by  six- 
teen.    Divide  the  result  by  seven. 

28.  Divide  two  hundred  sixteen  by  thirty-six  thousandths. 
Take  seventy -five  hundredths  from  the  quotient. 

29.  If  one  acre  yields  14  bu.  3  pk.  cranberries,  how  much 
will  40  acres  yield  ? 

30.  Find  the  difference  between  3£  x  6f  and  7^  -5-  If. 

31.  An  errand  boy  receives  $2.75  per  week.  In  how 
many  weeks  will  he  earn  enough  to  buy  a  pair  of  boots 
worth  $3.25,  a  coat  worth  $4.75,  a  hat  worth  $1.50,  and  6 
handkerchiefs  worth  25  cents  each? 

32.  How  many  cords  of  wood  at  $5J  a  cord  must  I  give 
for  78f  bushels  of  wheat  at  $1.20  a  bushel,  and  84  bushels 
of  rye  at  $1  a  bushel  ? 

33.  Mr.  Louis  Scott  bought  from  Thomas  Green,  at  Phil- 
adelphia, Jan.  10, 1904,  the  following :  67  pairs  of  boots  at 
$3.25  per  pair;  75  pairs  of  gaiters  at  $1.12  per  pair;  35 
pairs  of  slippers  at  70  cents  per  pair ;  50  pairs  of  rubbers  at 
62£  cents  per  pair.     Make  out  and  receipt  the  bill. 

34.  What  will  £  of  a  yard  of  cloth  cost,  if  £  of  a  yard 
costs  $1.60  ? 

35.  Divisor  3£;  quotient  400.     Find  dividend. 

36.  Dividend  .014 ;  quotient  2000.     Find  divisor. 

37.  Divide  118.35  by  .04£,  and  add  3.0045  to  the  quotient. 

38.  If  If  yards  of  cloth  are  worth  11  \  dollars,  what  is  a 
yard  worth  ? 


1 68  Chapter  Three. 

39.  If  a  roll  of  carpet,  containing  75  yards,  is  worth 
$  132,  what  is  f  of  a  yard  worth  ? 

40.  How  many  quarts  of  berries  at  11  cents  a  quart  will 
it  take  to  buy  2|  yards  of  cloth  at  16J  cents  a  yard  ? 

41.  A  man  sold  -J-  and  ^  of  his  farm  and  had  26 1  acres 
left.     How  many  acres  had  he  at  first  ? 

42.  A  boy  sleeps  |  of  his  time,  plays  -J-  of  it,  and  goes 
to  school  one-half  the  remainder.  How  many  hours  is  he 
in  school  each  school  day  ? 

43.  Write  in  four  other  ways  the  quantity  or  value  ex- 
pressed by  .16. 

44.  Bought  3  bu.  2  pk.  of  oats  for  $  1.38  and  retailed 
them  at  $  .12|  a  peck.     What  was  the  gain  ? 

45.  From  a  hogshead  of  molasses  containing  54  gal.  2  qt. 
there  was  sold  23  gal.  1  pt.  What  was  the  value  of  the 
remainder  at  8  cents  a  quart  ? 

46.  What  is  the  result,  if  the  sum  of  5  yd.  2  ft.,  3  yd. 
1  ft.,  and  14  yd.  1  ft.  be  taken  from  42  yards  ? 

47.  Eeduce  £  of  a  day,  ^j-  of  an  hour,  and  T47  of  a  minute 
to  common  denominator,  and  add. 

48.  Bought  a  carriage  for  $180,  and  after  paying  10% 
for  repairs,  sold  it  at  a  profit  of  25%  of  the  total  cost. 
Find  gain  and  selling  price. 

49.  A  man  sold  a  horse  for  $  125,  and  received  in  pay- 
ment 12£  yards  of  cloth  at  $  3.25  a  yard,  and  the  balance 
in  tea  at  $  .62 \.     How  many  pounds  of  tea  did  he  receive  ? 

50.  Find  equivalent  per  cents  for  the  following :  £,  |,  -J, 

h  &,  f  • 

51.  If  64  tons  of  iron  cost  $4816,  how  many  tons  can 
be  bought  for  $  1730.75  ? 

52.  Change  28  gal.  3  qt.  to  quarts. 


Review.  1 69 

53.  A  man  carried  to  a  store  75f  bushels  of  potatoes,  and 
received  for  them  27-J^  a  bushel.  How  many  yards  of  cloth, 
at  17f  ^  a  yard,  would  have  paid  for  them  ? 

54.  What  will  75  men  earn  in  18f  days,  if  each  earns  2\ 
dollars  each  day  ? 

55.  What  will  8  yd.  2  ft.  6  in.  of  silver  wire  cost  at  8f t 
an  inch  ? 

56.  A  young  man  spent  $195^  during  his  first  term  at 
college,  which  was  f  of  his  year's  allowance.  What  was  his 
year's  allowance,  and  what  had  he  left  for  the  remainder  of 
the  year  ? 

57.  A  man  paid  $  18.60  for  a  load  of  hay  weighing 
2 1  tons.  At  the  same  rate,  what  should  he  pay  for  \  of 
a  ton? 

58.  Divide  4.5006  by  .015. 

59.  One  man  owns  -^j-  of  an  estate ;  another  owns  f|-f 
of  it ;  and  a  third  man  owns  -£^  of  it.  What  part  of  the 
whole  do  they  own  together  ? 

Note.  — Reduce  the  fractions  to  lowest  terms,  by  inspection. 

60.  If  it  takes  11  men  45§  days  to  do  a  piece  of  work, 
how  many  days  will  it  take  one  man  to  do  the  same  work  ? 

61.  I  owned  f  of  a  house,  and  sold  f  of  my  share  for 
$1750.  What  was  the  value  of  the  whole  house  at  that 
rate? 

62.  A  grocer,  after  selling  -J-,  f ,  /¥,  and  \  of  a  quantity  of 
sugar,  had  102  pounds  left.  How  many  pounds  did  he  have 
at  first  ? 

63.  A  dealer  in  grain  bought  wheat  at  94^  a  bushel  to 
the  amount  of  $59.22,  and  sold  it  for  $70.56.  What  was 
the  selling  price  per  bushel  ? 

64.  If  I  of  a  cord  of  wood  is  worth  $  3.75,  what  will  J  of 
a  cord  cost  ? 


170  Chapter  Three. 

65.  A  man  who  had  $  50 J-,  received  $  S\  more,  spent 
$  17},  lost  $  4y%,  and  collected  $  15£  of  a  debt  How  much 
money  had  he  then  ? 

66.  12f  is  what  part  of  29  ? 

67.  What  must  a  carpenter  pay  for  the  following :  6500 
shingles,  at  $  4.75  per  thousand ;  15,964  feet  of  boards,  at 
$  39.25  per  thousand ;  4849  feet  of  planks,  at  $  45.32  per 
thousand  ? 

68.  A  farmer  sold  -§  of  his  wheat  for  $  796|  and  received 
for  it  $  ly1^  per  bushel.  How  many  bushels  did  he  have  at 
first,  and  how  many  did  he  sell  ? 

69.  If  123  tons  of  coal  cost  $  848.70,  what  will  be  the 
cost  of  265  tons  ? 

70.  A  dealer  sold  -^  of  his  wheat  to  Mr.  Adams,  -J  of  it 
to  Mr.  Baker,  and  y%  of  it  to  Mr.  Charles ;  then  he  had  630 
bushels  left.     How  much  had  he  at  first  ? 

71.  Mr.  Blank  bottled  135  gallons  of  ink  in  bottles  that 
held  I  of  a  pint ;  he  sold  it  for  \2\$  a  bottle.  How  much 
did  he  receive? 

72.  Three  times  a  number,  increased  by  -^  of  the 
number,  equals  22.      What  is  the  number? 

73.  A  grocer  having  a  capital  of  $  10,000,  invested  \  of 
it  in  tea  at  -fa  of  a  dollar  per  pound,  ^  of  the  remainder  in 
coffee  at  \  of  a  dollar  a  pound,  and  -fe  of  the  rest  in  sugar 
at  5  cents  per  pound.  What  quantity  of  each  did  he  buy, 
and  what  money  had  he  left  ? 

74.  What  will  be  the  cost  of  53,715  pounds  of  wheat  at 
90  cents  per  bushel  of  60  pounds  ? 

75.  A  drover  sold  15  cattle,  weighing  1468  pounds  each, 
at  $  4.40  per  hundred  pounds.     How  much  did  he  receive  ? 

76.  After  losing  |  of  his  money,  a  man  had  $  75  left 
How  much  had  he  at  first? 


Review.  171 

77.  What  will  be  the  cost  of  24  gallons  3  quarts  of  milk 
at  4  cents  per  pint  ? 

78.  A  man  bought  a  house  for  $6250  and  sold  it  for 
$6500.  What  fraction  of  the  cost  is  the  profit?  What 
decimal  ? 

79.  At  $30  per  month,  how  much  rent  would  a  man  pay 
from  July  1,  1904,  to  May  1,  1906? 

80.  How  many  sheep  at  $6.75  each  should  be  given  in 
exchange  for  54  horses  worth  $  160  each  ? 

81.  A  man  spent  three-tenths  of  his  money  for  clothes, 
and  one-fifth  of  it  for  rent,  and  had  $  75  left.  How  much 
did  his  clothes  cost? 

82.  What  would  be  the  cost  of  48,500  stamped  envelopes 
at  $21.30  per  thousand  ? 

83.  The  width  of  a  room  is  £  of  its  length.  How  many 
square  feet  in  the  floor,  if  the  width  is 

15  feet  ? 

84.  If  2  lb.  6  oz.  of  tea  cost  95  cents,  '/*» 
how  many   pounds   and   ounces  can  be  -- 
bought  for  $2.35? 

85.  John  and  James  went  out  together,  John  had  38 
cents.  When  one  of  the  boys  had  spent  18  cents  and  the 
other  had  spent  16  cents,  they  had  24  cents  left  between 
them.     Find  the  amount  of  money  James  had. 

86.  Find  \  of  the  sum  of  f  and  f . 

87.  What  is  \  of  the  difference  between  $  and  -|? 

88.  What  fraction  added  to  f  gives  f  ? 

89.  Change  1^-  hour  to  seconds. 

90.  £  of  what  number  equals  180  ? 

21a 

91.  The  half   of   a  number   added  to  its  - 

fourth  part  equals  21f.     What  is  the  number? 


172  Chapter  Three. 

92.  A  farm  is  sold  for  $5700,  at  a  loss  of  -fa  of  the  cost. 
What  was  the  cost  ? 

93.  When  it  is  noon  at  Philadelphia,  it  is  15  seconds  and 
10  minutes  past  5  p.m.  at  Paris.  What  time  is  it  at  Phila- 
delphia when  it  is  noon  at  Paris  ? 

94.  A,  B,  and  C  buy  a  house. 

A  furnished  }  of  the  cost,  B  fc  1  A         1      B      I  C    1 

and  C$1200.     What  did  A  and  *  *       $1200° 

B  pay,  respectively  ? 

95.  After  James  has  spent  f  of  his  money  and  \  of  the 
remainder,  he  has  but  $  1.50  left.  How  much  had  he  at 
first? 

96.  A  man  buys  oranges  at  $1.20  per  100.  How  many 
would  he  have  to  sell,  at  25#  per  dozen,  to  gain  $3.18? 

97.  From  a  piece  of  cloth  measuring  28 J  yards,  there 
have  been  sold  2|  yards,  6|  yards,  13}  yards.  If  the  re- 
mainder is  worth  $13.10,  what  was  the  value  of  the  whole 
piece  ? 

98.  A  man  left  for  charitable  purposes  $3600,  which  was 
I  of  his  money.  The  remainder  was  divided  equally  among 
8  relatives.     How  much  did  each  relative  receive  ? 

$3600. 


k  %  i 

Charitable  purposes 


tit 

Eight  relatives 


CHAPTER  IV. 

PAGES 

Denominate  Numbers 173  to  189 

Reduction,  Descending  and  Ascending,  Compound 
Addition,  Subtraction,  Multiplication,  and  Division, 
Avoirdupois  Weight,  Time  between  Dates. 

Percentage 189  to  194 

Applications  and  Simple  Interest. 

Measurements 195  to  209 

Area  of  Rectangles,  Square  Measure,  Solid  Contents, 
Cubic  Measure,  Surfaces  of  Rectangular  Solids, 
Angles,  Triangles,  Quadrilaterals. 

Review  of  Simple  Numbers  and  Fractions         .        .     209  to  218 
Special  Drills,  Sight  Approximations,  Fundamental 
Processes,    Cancellation,   Review  Fractions,   Review 
Decimals. 

Review  Problems .         .     218  to  228 

Miscellaneous,  Oral,  Written. 

DENOMINATE  NUMBERS. 

249.  Preliminary  Exercises. 

How  many  quarts  in  5  gal.  ? 
How  many  quarts  in  5  gal.  3  qt.  ? 
How  many  pints  in  23  qt.  ? 
How  many  pints  in  23  qt.  1  pt.  ? 
How  many  pints  in  5  gal.  3  qt.  1  pt.  ? 

REDUCTION  DESCENDING. 

250.  Keduce  5  gal.  3  qt.  1  pt.  to  pints. 

In  the  first  few  examples,  write  4  (the  number  of  quarts  in  a  gallon) 
above  the  quarts,  and  2  (the  number  of  pints  in  a  quart)  above  the 

173 


174  Chapter  Four. 

pints.     In  5  gallons  there  are  5  times  4  quarts,  or  20  quarts ;  adding 
the  3  quarts,   we  have  23  quarts,    as  the 

equivalent  of  5  gallons  3  quarts,  which  is  4  qt.    2  pt. 

written    in  the  column   of  quarts.      In  23  5  „a^     3  qt.     1  pt. 

quarts 'there  are  23  times  2  pints ;   adding  1  ~          ._     ~~ 

pint,  we  have  47  pints  as  the  equivalent  of  5  ^         ' 

gallons  3  quarts  1  pint.     This  is  written  in  the  Ans.     47  pt. 
column  of  pints,  the  23  quarts  being  cancelled. 

Changing  a  denominate  number  to  an  equivalent  denominate 
number  of  a  lower  denomination  is  called  reduction  descending. 


251.  Written  Exercises. 
Reduce  to  pints : 

1.  16  gal.  1  qt.  1  pt.  6.  31 J  gal. 

2.  27  gal.  2  qt.  7.9  gal.  2 J  qt. 

3.  16  gal.  8.  10  gal.  2  qt.  1  pt. 

4.  16  gal.  1  pt.  9.  27  gal.  1  pt. 

5.  34  gal.  3  qt.  1  pt.  10.  4  gal.  3  qt.  1£  pt. 

REDUCTION  ASCENDING. 

252.  Change  67  pt.  to  gallons,  quarts,  and  pints. 

Place  2  (the  number  of  pints  in  a  quart)  above  67  pints.    In  67  pints 

there  are  33  quarts  and  1  pint.     Write 

33  quarts  to  the  left  of  67  pints,  and  4  qt.    2  pt. 

the  1  pint  remainder  in  the  column  of  03  qt   qj  p^ 

pints.     Change  the  33  quarts  to  8  gal-      ~       r-:     ;      :     7       A 
%      *  *  ,00  8  gal.  1  qt.    1  pt.     Ans. 

Ions  1  quart,  and  cancel  33  quarts.  *  * 

Changing  a  denominate  number  to  an  equivalent  denominate 
number  of  a  higher  denomination  is  called  reduction  ascending. 


Denominate  Numbers.  175 

253.  Written  Exercises. 
Change  to  gallons,  etc. 

1.  156  qt.  6.  177  pt. 

2.  79  qt.  7.  139  pt. 

3.  408  pt.  8.  171  qt. 

4.  1302  pt.  9.  63  qt. 

5.  63  pt.  10.  711  pt. 

254.  Review  the  tables  of  Long  Measure,  Dry  Measure,  Liquid 
Measure,  Avoirdupois  Weight,  and  Time,  Art.  93,  pages  43-44. 

Change : 

1.  17  yd.  1  ft.  9  in.  to  inches. 

2.  4  mi.  100  rd.  4  yd.  to  yards. 

3.  74  bu.  2  pk.  7  qt.  to  quarts. 

4.  156  lb.  11  oz.  to  ounces. 

5.  63  yd.  0  ft.  3  in.  to  inches. 

6.  19  bu.  0  pk.  3  qt.  to  quarts. 

7.  11  rd.  3£  yd.  to  feet. 

8.  63  gal.  3  qt.  to  pints. 

9.  3  bu.  6  qt.  to  quarts. 

10.  17  T.  369  lb.  to  pounds. 

11.  15  hr.  16  min.  to  seconds. 

12.  4  wk.  6  da.  11  hr.  to  hours. 

Note.  —  Reduce  a  denominate  fraction  or  a  denominate  decimal  to 
lower  denominations  by  multiplying. 

13.  -f-  of  a  week  to  hours. 

14.  ■£%  of  a  mile  to  yards. 

15.  .00125  ton  to  ounces. 

16.  1876  inches  to  yards,  etc. 

17.  475  ounces  to  pounds,  etc. 


176  Chapter  Four. 

18.  729  quarts  to  bushels,  etc. 

19.  8675  minutes  to  days,  etc. 

20.  4972  pounds  to  tons,  etc. 

21.  972  rods  to  miles,  etc. 

22.  117  pints  to  gallons,  etc. 

23.  9483  seconds  to  hours,  etc. 

24.  877  quarts  to  bushels,  etc. 

25.  1495  ounces  to  pounds,  etc. 

26.  373  inches  to  yards,  etc. 

27.  216  quarts  to  gallons,  etc. 

28.  876  rods  to  miles,  etc. 

29.  319  pints  to  gallons,  etc. 

30.  3520  yards  to  miles. 

255.   Oral  Exercises. 

1.  How  many  hours  in  §  of  a  day  ? 

2.  How  many  hours  in  -^  of  a  day  ? 

3.  How  many  minutes  in  £  of  an  hour  ? 

4.  How  many  hours  and  minutes  in  £  of  a  day  ? 

■J  day  ss  4$  hours  ;    f  hour  =  48  minutes.     \  day  =  4  hours  48 
ninutes. 

5.  How  many  quarts  and  pints  in  f  of  a  gallon  ? 

6.  How  many  hours  and  minutes  in  .2  day  ? 

.2  day  =  4.8  hours ;    .8  hour  =  48  minutes.     .2  day  =  4  hours  48 
minutes. 

7.  How  many  quarts  and  pints  in  .375  gallon  ? 

8.  Change  .3  day  to  hours  and  minutes. 

9.  Change  .625  bushel  to  pecks  and  quarts. 

10.  What  part  of  a  gallon  is  1  pint  ? 

11.  What  part  of  a  gallon  is  3  pints  ? 


Denominate  Numbers.  177 

12.  What  part  of  a  gallon  is  1  qt.  1  pt.  ? 

13.  What  decimal  of  a  gallon  is  1  qt.  1  pt.  ? 

14.  What  decimal  of  a  gallon  is  2  qt.  1  pt.  ? 

15.  What  part  of  2  gallons  is  2  qt.  1  pt.  ? 

16.  Change  .375  bushel  to  pecks  and  quarts. 

17.  What  decimal  of  a  bushel  is  4  quarts  ? 

18.  What  fraction  of  a  day  is  3  hr.  20  min.  ? 

19.  Eeduce  960  minutes  to  hours. 

20.  How  many  minutes  in  a  day  ? 

256.   Written  Exercises. 

1.  What  decimal  of  a  ton  is  3  pounds  ? 

Pounds  are  changed  to  tons  by  dividing  by  2000. 
3  lb.  =  jfa  T.  s  .0015  T.    Ans. 

2.  What  fraction  of  an  hour  is  12  min.  30  sec.  ? 

12  min.  30  sec.  =  12  J  min.  =^ihr.  =  M,  hr.  =  &. 

60 

3.  Eeduce  -£%  of  a  day  to  minutes. 

ft  day  =  (3V  x  24)  hr.  =  (^  X*£  X  ^)  min.    Cancel. 

4.  Eeduce  .03125  day  to  minutes. 

5.  What  decimal  of  a  day  is  9  minutes? 

6.  What  will  be  the  cost  of  15  T.  500  lb.  coal  at  $  7  per 
ton? 

7.  When  coal  is  $5  per  ton,  how  many  tons  and  pounds 
can  be  bought  for  $  18.75  ? 

8.  Change  2  ft.  7  in.  to  the  fraction  of  a  yard. 

2  ft.  7  in.  =  2TV  ft.  =  ?i  yd.  =  ft  yd. ,   Ans. 

Note.  —  An  expression  such  as  -J3  is  called  a  complex  fraction.     It 

o 

indicates  the  division  of  2^  by  3  ;  that  is,  f  \  x  |,  or  %\. 


178  Chapter  Four. 

9.   Eeduce  3  pk.  4  qt.  to  the  decimal  of  a  bushel. 

4  qt.  =  .5  pk. ;  3  pk.  4  qt.  =  3.5  pk.  =  —  bu. ;  etc. 

4 

10.  How  many  pecks  and  quarts  in  .9375  bushel  ? 

11.  If  .1875  of  a  gallon  of  cologne  cost  $  1.125,  what  will 

1  pint  cost? 

Note. — $.125  is  read  12  cents  5  mills. 

12.  Find  the  cost  of  42  gal.  3  qt.  1  pt.  oil,  at  16  cents 
per  gallon. 

13.  Reduce  ij  of  a  gallon  to  quarts  and  pints. 

14.  What  part  of  3  T.  is  1  T.  960  lb.  ? 

15.  A  man  raised  194  bu.  1  pk.  of  rye.     He  sold  129  bu. 

2  pk.     What  fraction  of  his  crop  did  he  sell  ? 

16.  10  bu.  1  pk.  of  seed  are  packed  in  8  bags.     What 
quantity  is  there  in  each  bag  ? 

17.  What  decimal  of  a  day  is  15  hr.  45  min.  ? 

18.  How  many  feet  are  there  in  a  mile  ? 


COMPOUND  ADDITION. 

A  compound  denominate  number  expresses  two  or  more  denomina- 
tions of  the  same  kind. 

316  T.  1816  lb.  is  a  compound  denominate  number. 
487  T.  is  a  simple  denominate  number. 

In  adding  and  subtracting  compound  denominate  numbers, 
write  units  of  the  same  denomination  in  the  same  column. 

257.   Add  the  following : 

1.        18  bu.  3  pk.  7  qt.  7  qt.  +  4  qt.  +  6  qt.  =  17  qt.  =  2  pk.  1  qt. 

9  bu.  2  pk.  4  qt.  Write  *  qt.  and  carry  2  pk.    2  pk.  +  2  pk. 

14  bu.  1  pk.  6  qt.  +  1  Pk-  +  2pk.+  3pk.  =  lOpk.  =  2bu.2pk. 

2  pk  write  2  pk.  and  carry  2  bu.     2  bu.  +  14  bu. 


Ans.  43  bu.  2  pk.  1  qt. 


f  9  bu.  +  18  bu.  =  43  bu. 


Denominate  Numbers. 

16  yd.  2  ft.    9  in.  6.   12  T.  1576  lb. 

17  yd.            4  in.  3  T.    980  1b. 

lft.    6  in.  4761b. 

11  in.  1  T.  1830  lb. 


19 


11  da.    5  hr.  19  min.  7.    2  wk.  5  da.  12  hr. 

23  da.             40  min.  6  da.  15  hr. 

17  hr.  50  min.  5  wk.              2  hr. 

5  da.  20  hr.    6  min.  2  da.  19  hr. 


4.  93  gal.  3  qt.  1  pt.  8.  18  mi.  100  rd. 
74  gal.  34  rd. 
18  gal.  1  qt.  29  mi. 

2  qt.  1  pt.  6  mi.  160  rd. 

5.  5  hr.  30  min.  20  sec.  9.    47  yr.  11  mo. 

45  min.  33  sec.  5  yr.    9  mo. 

6  hr.  11  min.    5  sec.  7  mo. 

10  hr.    3  min.  30  sec.  22  yr.    5  mo. 

10.  487  T.,  316  T.  1816  lb.,  247  lb.,  43  T.  811  lb.,  19  T.  25  lb. 

11.  83  lb.  15  oz.,  9  lb.  5  oz.,  18  lb.,  22  lb.  11  oz.,  5  lb.  8  oz. 

12.  8  hr.  15  min.  5  sec,  37  min.  52  sec,  5  hr.  48  min.,  23  hr. 

13.  72  gal.  3  qt.  1  pt.,  17  gal.  1  qt.,  2  qt.  1  pt.,  90  gal.  1  pt. 

14.  7  yd.  2  ft.  11  in.,  19  yd.  6  in.,  105  yd.,  4  yd.  2  ft.  2  in.,  1  ft. 

15.  93  mi.  300  rd.,  87  mi.  154  rd.,  194  rd.,  3  mi.  175  rd.,  9  mi. 

COMPOUND   SUBTRACTION. 

258.    Subtract: 

Change  83  yr.  3  mo.  to  82  yr.  15  mo.    Sub- 
1.    83  yr.  3  mo.      tract  9  months  from  15  months.     Write  the 
15  yr.  9  mo.      remainder,  6  months,  in  the  column  of  months. 
Subtract  15  years  from  82  years. 

Ans.  67  yr.  6  mo. 


180  Chapter  Four. 

2.    62  mi.    84  rd.  7.    18  hr.  5  min. 

19  mi.  159  rd.  40  min.  25  sec. 


3. 

76  T.  225  lb. 
37  T.  1679  lb. 

4. 

100  lb. 
83  lb.  4  oz. 

5. 

52  wk. 

13  wk.  3  da.  7  hr. 

6. 

19  gal.     1  pt. 
8  gal.  3  qt. 

8. 

16  yd.      9  in. 
7  yd.  1  ft.  11  in. 

9. 

100  bu. 
42  bu.  3  pk.  7  qt. 

10. 

45  da.  1  hr.  1  min. 
6  da.  6  hr.  6  min. 

11.  From  27  bu.  1  pk.  5  qt.  take  13  bu.  3  pk.  7  qt. 

12.  From  100  gal.  1  qt.  take  83  gal.  2  qt.  1  pt. 

13.  From  22  hr.  15  min.  20  sec.  take  15  hr.  45  min.  40  sec. 

14.  From  17  lb.  2  oz.  take  13  lb.  8  oz. 

15.  From  100  bu.  take  74  bu.  2  pk.  1  qt. 

COMPOUND   MULTIPLICATION. 

259.   Written  Exercises. 

Multiply  4  gal.  3  qt.  1  pt.  by  3. 

3  times  1  pt.  =  3  pt.     3  pt  =  1  qt. 

4  gal.  3  qt.  1  pt.  1  pt.     Write  1  pint  in  the  column  of 

3  pints.    3  times  3  qt.  =  9  qt. ;  9  qt.  + 


14  gal.  2  qt.  1  pt.  Ans.       the  1  <$• t0  cari7  =  10  %*>•    10  <$-  = 

2  gal.  2  qt.     Write  2  quarts  in  the 
column  of  quarts.     3  times  4  gal.  = 
12  gal. ;  12  gal.  +  2  gal.  to  carry  =  14  gal.     Ans.  14  gal.  2  qt.  1  pt. 


Denominate  Numbers.  181 

Multiply : 

1.  13  bu.  3  pk.  6  qt.  by  2.  7.  25  lb.  4  oz.  by  8. 

2.  25  gal.  2  qt.  1  pt.  by  3.  8.  33  min.  33  sec.  by  9. 

3.  7  lb.  10  oz.  by  4.  9.  2  pk.  7  qt.  by  10. 

4.  23  bu.  3  qt.  by  6.  10.  3  qt.  1  pt.  by  11. 

5.  32  gal.  1  pt.  by  7.  11.  4  yr.  6  mo.  by  12. 

6.  3    hr.  15    min.  15  sec.         12.   5   wk.    6    da.   12   hr. 
by  5.  by  16. 

COMPOUND  DIVISION. 

260.   Divide  54  yd.  1  ft.  4  in.  by.  20. 

54  yd.  -T-  20  gives  a  quotient  of  2  yd., 

2 — 2_j : IB;  which  is  written,  and  a  remainder  oi 

2  yd.     2  ft.     2  m.  Ans.       «.      .      _   .        '  .   .     Aa '  , 

14  yd.     Reduce  14  yd.  to  42  ft.,  and 

add  1  ft.,  making  43  ft.     43  ft.  —  20  gives  a  quotient  of  2  ft.,  which  is 

written,  and  a  remainder  of  3  ft.    Reduce  3  ft.  to  36  in.,  and  add  4  in., 

making  40  in.    40  in.  -f-  20  gives  a  quotient  of  2  in.  which  is  written. 

Divide : 

1.  13  wk.  by  5.  7.  17  lb.  7  oz.  by  3. 

2.  15  lb.  9  oz.  by  3.  8.  37  bu.  3  pk.  6  qt.  by  2. 

3.  2  lb.  3  oz.  by  5.  9.  67  yd.  2  ft.  by  4. 

4.  2  gal.  1  qt.  by  3.  10.  33  da.  15  hr.  57  min.  by  3. 

5.  5  bu.  by  4.  11.  561  gal.  by  6. 

6.  7  hr.  by  6.  12.  22  hr.  20  min.  20  sec.  by  4. 

13.  109  gal.  1  qt.  1  pt.  by  7. 

14.  273  yd.  1  ft.  6  in.  by  9. 

15.  155  bu.  3  pk.  2  qt.  by  6. 

16.  180  da.  19  hr.  28  min.  by  8. 


i8* 


Chapter  Four. 


17.   Divide  243  da.  4  hr.  2  min.  by  15. 
Dividing    243    days 


15)243  da. 

,  15 

93  da. 

_90  da. 

3  da. 


by  15  gives  a  quotient 
of  16  days  and  a  re- 
mainder of  3  days.   Re- 
ducing 3  days  4  hours 
to  76  hours  and  divid- 
ing by  15  gives  a  quo- 
tient of  5  hours  and  a 
remainder  of   1   hour. 
Reducing  1  hour  2  minutes  to  62 
minutes  and  dividing  by  15  gives 
a  quotient  of  4  minutes  and  a 
remainder  of  2  minutes.     Reducing  2  min- 
utes to  120  seconds  and  dividing  by  15  gives 
a  quotient  of  8  seconds. 

18.  Divide  334  yd.  9  in.  by  21. 
15  yd.  2  ft.   9  in. 


16  da.  5  hr.  4  min.   8  sec. 


4  hr.  2  min. 


4hr. 


76  hr. 

75  hr. 

lhr. 


2  min. 


62  min. 

60  min. 

2  min. 


120  sec. 
120  sec. 


21)334  yd.  0  ft. 
21 

124  yd. 

105  yd. 

19  yd. 

9in. 

57  ft. 

42  ft. 

15  ft. 

9  in. 

189  in. 

189  in. 

19. 

825  lb.  by  48. 

20. 

112  T.  by  25. 

21. 

43  mi.  by  32. 

22. 

84  yr.  by  24. 

23. 

462  bu.  by  32. 

24. 

1078  yd.  by  62 

i 

Insert  the  missing  denomina- 
tion, feet,  with  a  cipher  prefixed. 
Reduce  the  19  yards  remainder  to 
57  feet.  Reduce  to  189  inches  the 
15  feet  9  inches  remaining. 


25.  288  hr.  9  min.  by  54. 

26.  863  gal.  2  qt.  1  pt.  by  47. 

27.  33  wk.  1  da.  by  72. 

28.  1138  T.  910  lb.  by  81. 

29.  1629  yd.  1  ft.  by  96. 

30.  1867  gal.  1£  pt.  by  125. 


Denominate  Numbers.  183 

261.   Avoirdupois  Weight.     Long  Ton. 

In  selling  iron,  coal  at  the  mines,  ores,  etc.,  and  in  calculating  the 
duties  at  the  U.  S.  custom  houses  upon  imported  goods,  the  following 
table  is  used : 

28  pounds  (lb.)      =  1  quarter  (qr.) 

4  quarters  =  1  hundredweight  (cwt.) 

20  hundredweight  =  1  ton  (T.) 


1  cwt.  =  112  lb.        1  T.  =  2240  lb. 

The  ton  of  2240  pounds  is  called  a  long  ton.  Unless  otherwise 
specified  in  a  problem,  the  cwt.  of  100  pounds  and  the  ton  of  2000 
pounds  are  to  be  taken. 

262.    Oral  Problems. 

1.  How  many  tons  and  pounds  of  coal  in  40  bags,  each 
containing  80  pounds  ? 

2.  If  it  takes  3  hr.  20  min.  to  hoe  a  row  of  corn,  how 
long  will  it  take  to  hoe  3  rows  ? 

3.  A  man  puts  up  3^-  pounds  of  tea  into  4  ounce  pack- 
ages.    How  many  packages  does  he  make  ? 

4.  3  pk.  3  qt.  of  apples  are  divided  among  9  children. 
What  quantity  does  each  child  receive  ? 

5.  What  part  of  a  day  is  30  minutes  ? 

6.  If  there  are  2\  gallons  of  wine  in  12  bottles,  how 
many  pints  are  there  in  each  bottle  ? 

7.  What  is  the  weight  of  two  packages  each  containing 
15  lb.  11  oz.  ? 

8.  What  part  of  an  hour  is  40  seconds  ? 

9.  What  is  the  rent  of  a  house  for  1  year  9  months  at 
$16  per  month? 

10.  If  3  gal.  2  qt.  1  pt.  of  milk  are  taken  from  a  can  con- 
taining 10  gallons,  how  much  is  left  in  the  can  ? 


184  Chapter  Four. 

11.  5  hams  weigh  61 J  pounds.    What  is  the  average 
weight  ? 

12.  There  are  on  an  average  41  pupils  in  a  class.     How 
many  are  there  in  14  classes  ? 

13.  At  37^-  cents   per  yard,  how  many  yards   can  be 
bought  for  $6.75? 

•  6f-*.$t  =  ¥  +  t  =  ¥-*-t.  etc. 

14.  Find  the  cost  of  16  barrels  of  flour  at  $6.12£  each. 

15.  $1.65  is  equally  divided  among  15  boys.     What  is 
the  share  of  each  ? 

16.  A  floor  containing  40 J  square  yards  is  7  yards  long. 
How  many  yards  wide  is  it  ? 

17.  How  many  ounces  in  5J  pounds  ? 

263.   Written  Problems. 

1.  If  a  watch  gains  1  min.  17  sec.  per  day,  how  much 
will  it  gain  during  March  and  April  ? 

2.  How  many  bushels,  pecks,  and  quarts  in  1449  pounds 
of  corn,  weighing  56  pounds  to  the  bushel  ? 

3.  Eeduce  25  T.  13  cwt.  2  qr.  25  lb.  to  pounds  (long 
ton). 

4.  A  chain,  97  yd.  8  in.  long,  contains  1000  links.  Find 
the  length  of  one  of  the  links. 

5.  A  farmer  sold  out  of  5  bushels  of  peas  the  following 
quantities :  3  pk.  6  qt. ;  4  pk. ;  4  pk.  3  qt. ;  1  bu.  1  pk.  1  qt. 
How  much  has  he  still  to  sell  ? 

6.  Change  100,000  pounds  to  tons  (long),  cwt.,  qr.,  lb. 

7.  A  man  walks  on  Monday  15  mi.  161  rd. ;  Tuesday, 
10  mi.  84  rd. ;  Wednesday,  19  mi.  15  rd. ;  Thursday  and 
Friday,  12  mi.  121  rd.  each  day ;  Saturday,  14  mi.  240  rd. 
What  distance  per  day  does  he  average  ? 


Denominate  Numbers.  185 

8.  If  the  sun  rises  at  5  hr.  10  min.  a.m.,  and  sets  at  6  hr. 
42  min.  p.m.,  how  long  is  the  day  ?  How  many  hours  and 
minutes  of  night  ? 

9.  Find  the  duty  at  1-j2^  per  pound  on  an  invoice  of  tin 
weighing  33  T.  7  cwt.  20  lb.  (long  ton). 

10.  An  iron  rod  is  12  ft.  6  in.  long.  Prom  it  are  cut 
73  bolts,  each  If  inches  long.     How  much  is  left  ? 

11.  A  man  rows  a  mile  in  10  min.  30  sec.  How  long 
would  he  take  to  row  27  miles  at  the  same  rate  ? 

12.  What  is  the  total  weight  in  tons  (long),  etc.,  of  19 
barrels  of  sodarash  weighing  13  cwt.  2  qr.  10  lb.  each  ? 

13.  A  man  rows  51  miles  in  23  hr.  5  min.  and  30  sec. 
How  long  does  he  take  to  row  a  mile  ? 

14.  If  I  lost  $50  by  selling  a  lot  for  two-thirds  of  its 
cost,  what  would  I  have  lost  if  I  had  sold  it  for  three-fourths 
of  its  cost  ? 

15.  At  the  rate  of  $2.75  per  day  of  8  hours,  how  much 
should  be  given  a  man  that  works  from  a  quarter  before  8  in 
the  morning  until  5  minutes  past  11  in  the  morning  ? 

16.  If  a  railroad  train  travels  18  miles  in  40  minutes, 
how  far  will  it  travel,  at  the  same  rate,  in  1\  hours  ? 

17.  A  coal  dealer  buys  175  (long)  tons  of  coal.  How 
much  does  he  receive  for  it  at  $  5  per  ton  of  2000  pounds  ? 

TIME  BETWEEN  DATES. 

264.   Oral  Problems. 

1.  How  many  hours  from  3  o'clock  Saturday  afternoon 
to  9  o'clock  Sunday  morning  ? 

2.  How  many  days  from  May  1  to  June  1  ? 

3.  A  boy  takes  a  spoonful  of  medicine  every  hour.  If 
he  takes  the  first  dose  at  2  o'clock,  at  what  hour  will  he  take 
the  sixth  ?     The  second  ?     The  fourth  ? 


1 86  Chapter  Four. 

4.  A  man  begins  work  on  the  morning  of  the  6th  and 
ends  on  the  evening  of  the  11th.  How  much  does  he  earn 
at  $  3  per  day  ? 

5.  An  importer  receives  some  cases  of  goods  numbered 
consecutively.  How  many  cases  are  there  if  the  lowest 
number  is  29  and  the  highest  number  is  53  ? 

6.  How  many  posts  6  feet  apart  will  be  needed  for  a 
fence  120  feet  long.  For  a  fence  6  feet  long  ?  12  feet 
long? 

7.  Find  the  time  from  Jan.  1  to  Jan.  31,  counting  the 
first  and  the  last  day.     Omitting  both  days. 

8.  How  many  days  from  July  4  to  Aug.  15,  inclusive  ? 

9.  How  many  chapters  from  the  25th  to  the  49th, 
exclusive  ? 

10.  A  girl  begins  at  the  146th  problem  and  solves  all 
those  on  two  pages.  If  the  last  is  the  172d  problem,  how 
many  does  she  solve  ? 

265.   How  many  days  from  March  4  to  Sept.  1  ? 

March  4  to  March  31,  27  days 

Excluding  March  4,  there  remain 
in  the  month  31  —  4,  or  27  days.  To 
this  add  the  number  of  days  in  April, 
May,  June,  July,  and  August.  Since 
March  4  is  excluded,  we  take  1  day 
in  September,  making  the  total  181 
days. 

In  finding  the  time  between  dates,  either  the  first  or  the 
last  day  is  excluded;  that  is,  from  the  1st  to  the  21st  is  con- 
sidered 20  days. 


April 

30 

May 

31 

June 

30 

July 

31 

Aug. 

31 

Sept. 

1 

Ans. 

181  days 

Denominate  Numbers.  187 

L   How  many  days  from 

11.  Jan.  1  to  Feb.  19?  16.  Feb.  29  to  April  1  ? 

12.  Oct.  31  to  Dec.  30  ?  17.  May  21  to  July  4  ? 

13.  Sept.  30  to  Dec.  16  ?  18.  April  7  to  May  27  ? 

14.  Nov.  1  to  Dec.  19  ?  19.  June  10  to  Aug.  1  ? 

15.  March  16  to  April  25  ?  20.  July  4  to  Sept.  1  ? 

267.   Written  Problems. 

Take  note  of  leap  year. 
How  many  days  from : 

1.  Feb.  6,  1903,  to  Oct.  1, 1903  ? 

2.  Oct.  14,  1903,  to  March  3,  1904? 

3.  Jan.  1,  1904,  to  April  19,  1904? 

4.  Dec.  23,  1904,  to  March  8,  1905  ? 

5.  Sept.  3, 1903,  to  Feb.  1, 1904  ? 

6.  March  16,  1904,  to  Dec.  25,  1904  ? 

7.  June  3,  1905,  to  Nov.  29,  1905  ? 

8.  Aug.  17,  1903,  to  Jan.  3,  1904  ? 

9.  April  4,  1905,  to  July  4,  1905  ? 
10.   May  16, 1906,  to  Oct.  14,  1906  ? 

11.  How  much  wages  at  $4  per  day  should  a  man  receive 
from  Tuesday,  Jan.  2,  1906,  to  Feb.  28,  inclusive,  no  pay  to 
be  received  for  Sundays  or  legal  holidays  ? 

12.  A  man  borrowed  $100  April  4,  and  returned  it  Nov. 
25.  How  many  days'  interest  did  he  owe  ?  (Do  not  include 
both  days.) 

13.  May  1,  1903,  fell  on  Friday.  Upon  what  day  of  the 
week  did  July  4  fall  ? 


1 88  Chapter  Four. 

14.  How  many  days  does  vacation  last  if  it  begins  on  the 
morning  of  Saturday,  July  2,  and  school  commences  on  the 
first  Tuesday  of  September  ? 

15.  A  man  borrows  some  money  June  16,  and  agrees  to 
return  it  in  60  days.     On  what  date  should  he  pay  it  ? 

16.  A  traveller  starts  upon  a  trip  Aug.  24,  1904,  and 
reaches  home  again  Feb.  10,  1905.     How  long  is  he  away  ? 

In  each  of  the  preceding  examples  the  difference  between  the  dates 
is  less  than  a  year,  and  the  answer  is  required  in  days.  When  the 
difference  is  more  than  a  year,  it  is  generally  obtained  by  compound 
subtraction,  each  month  being  considered  as  containing  30  days. 

17.  Find  the  difference  in  time  between  March  3,  1891, 
and  Jan.  1,1905.  ^        f         % 

Writing  1905,  1st  month,  1st  day,  we  subtract     1891  3  3 

1891,  3d  month,  3d  day.    Ans.  13  yr.  9  mo.  28  da.   

13         9       28 

18.  George  Washington  was  born  Feb.  22,  1732.  How 
old  was  he  at  the  signing  of  the  Declaration  of  Indepen- 
dence, July  4,  1776  ? 

19.  Abraham  Lincoln  was -first  inaugurated  president 
March  4,  1861.  How  long  had  he  served  at  his  death, 
April  15,  1865? 

20.  The  battle  of  Lexington  took  place  April  19,  1775. 
The  treaty  of  peace  was  signed  Sept.  3,  1783.  How  many 
years,  months,  and  days  between  the  two  events  ? 

21.  How  many  years  elapsed  between  the  discovery  of 
America  by  Columbus,  Oct.  12,  1492,  and  the  landing  of  the 
Pilgrims,  Dec.  21,  1620? 

22.  General  Harrison  fought  the  battle  of  Tippecanoe 
Nov.  7,  1811.  He  was  inaugurated  president  29  yr.  3  mo. 
27  da.  later.     Give  the  date  of  his  inauguration. 


Percentage. 


189 


23.  How  long  was  it  after  the  treaty  with  England, 
signed  Dec.  24,  1814,  that  the  Mexican  treaty  was  con- 
cluded, Feb.  2,  1848? 

24.  General  Taylor  died  July  9,  1850.  How  long  did  he 
live  after  the  capture  of  Monterey,  Sept.  24,  1846  ? 

25.  President  Garfield  was  born  Nov.  19,  1831.  How 
old  was  he  at  his  inauguration,  March  4,  1881? 

26.  The  last  battle  of  the  Mexican  War  took  place  Sept. 
14,  1847.  The  battle  of  Bull  Run  was  fought  13  yr.  10  mo. 
7  da.  later.    What  was  the  date  of  this  battle  ? 

27.  Find  the  time  between  July  4, 1776,  and  Jan.  1, 1904. 


PERCENTAGE 

268 

.   Oral  Exercises. 

1. 

Find  4%  of  $125. 

6. 

33  J  %  of  1  day. 

2. 

25%  of  16. 

7. 

62£%  of  $12. 

3. 

6%  of  200  cows. 

8. 

9  %  of  $23. 

4. 

1%  of  150  pounds. 

9. 

75  %  of  3  gallons. 

5. 

20%  of  65  yards. 

10. 

lJ%of  $400. 

269 

.  Written  Exercises. 

1. 

Find  6%  of  $576. 

9. 

25  %  of  $156. 

$576  x  .06 

£  of  $156 

2. 

41%  of  $340. 

10. 

1   %  of  $156. 

3. 

25  %  of  1876  bushels. 

11. 

i%  of  $156. 

4. 

121%  of  864  cows. 

12. 

50  %  of  480  hours, 

5. 

50  %  of  432  yards. 

13. 

\%  of  480  hours. 

6. 

33^%  of  576  soldiers. 

14. 

1%  of  $1420. 

7. 

16f  %  of  696  gallons. 

15. 

31%  of  $66. 

8. 

6J%of  $4.96. 

16. 

7|%  of  360  days. 

190  Chapter  Four. 

INTEREST. 

270.  Interest  is  the  sum  paid  for  the  use  of  money. 
The  Principal  is  the  sum  loaned. 

The  Amount  is  the  sum  of  the  principal  and  interest. 

In  computing  interest,  the  year  is  considered  as  composed 
of  12  months  of  30  days  each. 

271.  Oral  Exercises. 
Find  the  interest  on : 

1.  $90  for  2  months  at  4%. 

2.  $60  for  60  days  at  6%. 

3.  $100  for  2  yr.  6  mo.  at  5%. 

4.  $  120  for  30  days  at  5%. 

5.  $300  for  90  days  at  3%. 

6.  $100  for  1  yr.  3  mo.  at  4%. 

7.  $50  for  3  years  at  6%. 

8.  $100  for  2  yr.  4  mo.  at  6%. 

272.  Find  the  interest  on  $  63  for  4  yr.  5  mo.  at  5%. 

$63. 

.05 

Find  the  interest  for  one  year  by  multi-  $    3  15 

plying  the  principal,   $63,  by  the  rate,  6,  .  5 

expressed  as  hundredths.     Multiply  this  prod-  t-r 

uct,  $3.15,  by  the  time  expressed  in  years,  '? 

4&.  *    L31  + 


12.60 
Ans.  $  13.91 


$  63  is  called  the  principal. 

5  =  rate.  4  yr.  5  mo.  =  time. 

Rate 

Interest  =  Principal  x x  Time  (in  years). 

100 


Interest.  191 

The  work  may  sometimes  be  shortened  by  indicating  the 
operations  and  cancelling : 

4 

Find  the  interest  on  $160.50  for  3  mo.  15  da.  at  6%. 

$  mn  X  j|j  X  ?-  =  $114235  =  $  2.808  +     Ans.  $2.81. 

4 
Note.  —  The  divisor,  100,  should  be  cancelled  only  in  performing 
the  final  division. 

Find  the  interest  on  $69.75  for  1  mo.  17  da.  at  4%. 
$.007/75        s        47 

•  W-M  X  =gs  X  ^-  =  $  .36425.    Ans.  36  cents. 

Note.  —  The  three  ciphers  in  the  dividend  are  cancelled  by  moving 
the  decimal  point  in  the  dividend  three  places  to  the  left,  prefixing  a 
decimal  cipher. 

273.   "Written  Exercises. 
Find  the  interest  on : 

1.  $192  for  3  yr.  7  mo.  at  5%. 

2.  $  60  for  2  mo.  12  da.  at  4%. 

3.  $240  for  1  yr.  1  mo.  at  6%. 

4.  $14.40  for  5yr.  5  mo.  at  5%. 

5.  $36  for  77  days  at  41%. 

6.  $99  for  21  months  at  6%. 

7.  $  192  for  2  yr.  4  mo.  at  5%. 

8.  $600  from  Jan.  1  to  Jan.  16  at  4%. 

9.  $1200  from  July  1,  1903,  to  Jan.  1,  1905,  at  6%. 
10.    $57.60  from  Oct.  4,  1904,  to  Feb.  4,  1908,  at  5%. 


192  Chapter  Four. 

274.    Oral  Problems. 

1.  16  is  how  many  hundredths  of  64  ? 

2.  What  per  cent  of  25  is  5  ? 

3.  What  part  of  £  is  f  ? 

Change  both  to  the  same  denominator  :  16  twentieths,  15  twentieths. 

4.  What  part  of  2  lb.  1  oz.  is  1  lb.  ? 

Change  both  to  the  same  denomination  :  33  oz.,  16  oz. 

5.  Divide  4  gallons  by  3  pints. 

6.  How  many  pencils  at  4  mills  each  can  be  bought  for 
a  dollar  ?  !  mill  =  ^  0f  a  cent. 

7.  Write  ^asa  decimal. 

8.  Divide  34  by  200. 

9.  How  many  pounds  in  one-quarter  of  a  ton?     How 
many  pints  in  .25  of  a  bushel  ? 

10.  Change  37^,  75^,  8^,  62j£  6{f,  to  fractions  of  a 
dollar  ? 

11.  How  many  pounds  of  cheese  at  $0.16$  a  pound  can 
be  bought  for  $5.00? 

12.  An  agent  collected  rents  amounting  to  $300.     What 
was  his  commission  at  ^%  ? 

13.  Find  the  interest  of  $200  for  1  yr.  3  mo.  at  4%. 

14.  A  farmer  raised  50  bushels  of  cranberries,  and  sold 
60%  of  them.     How  many  bushels  did  he  sell  ? 

15.  What  %  of  a  number  is  fa  of  it  ? 

16.  What  would  42  pounds  of  butter   cost  at  33 \$  a 
pound? 

17.  When  the  tax  rate  is  $12  per  $1000,  what  will  Mr. 
Smith's  tax  be  if  he  owns  $4500  worth  of  property  ? 


Review.  193 

18.  A  man  pays  $60  interest  per  year.     How  much,  in- 
terest does  he  pay  in  3  yr.  7  mo.  ? 

19.  At  $45  per  month,  what  is  the  rent  of  a  house  for 
2  yr.  7  mo.  ? 

20.  Express  in  per  cents :  \ ;  \\  \ ;  \ ;  -J-. 

275.   Written  Problems. 

1.  What  is  the  interest  on  $760  for  5  months  at  3±%  ? 

2.  A  merchant  insures  property  worth  $20,000  for  J  of 
its  value.  How  much  does  he  pay,  the  rate  for  insuring 
being  1J%? 

3.  What  is  the  commission  on  $56*^8  worth  of  cloth  at 

4.  At  3%,  what  is  the  commission  on  the  sale  of  5000 
pounds  of  sugar  at  5^  per  pound  ? 

5.  What  will  be  the  interest  on  $720  for  3  mo.  24  da.  at 

6.  A  clerk's  income  is  $800.  He  pays  25%  of  it  for 
board,  and  33  J  %  of  the  remainder  for  clothes.  How  much, 
has  he  left  ? 

7.  \°lo  of  my  money  is  in  my  pocket,  38%  is  in  the  bank, 
and  the  rest  is  in  real  estate.  I  have  in  all  $  24,000.  How 
much  is  in  the  bank  and  in  real  estate  ? 

8.  An  auctioneer  sold  for  Mrs.  Paul,  on  10  %  commission, 
14  chairs  at  $1.75,  6  tables  at  $2.75,  40  yards  carpet  at 
62i^  a  yard,  and  a  miscellaneous  lot  for  $119.24.  What 
sum  did  Mrs.  Paul  receive  after  paying  commission  ? 

9.  How  many  feet  in  62^%  of  a  mile  ? 
What  part  of  a  day  is  18  hr.  30  min.? 
Reduce  9  cwt.  17  lb.  to  ounces. 

10.   If  .625  of  a  cord  of  wood  costs  $3.75,  what  will  .75 
of  a  cord  cost  ? 


194  Chapter  Four. 

11.  A  business  man's  receipts  for  a  week  are  $2575. 
His  average  rate  of  profit  is  5%  of  his  receipts.  What  is 
his  profit  for  the  week  ? 

12.  A  certain  city  had  14,250  inhabitants  in  1900.  The 
population  has  increased  24  per  cent.  What  is  the  present 
number  of  inhabitants  ? 

13.  A  class  has  56  pupils  on  register.  When  14^  per 
cent  of  the  pupils  are  absent,  how  many  are  present  ? 

14.  A  merchant's  sales  for  1903  were  $  45,276.  What 
should  be  the  sales  for  1904  to  make  an  increase  of  16f  per 
cent  ? 

15.  Thirty  words  were  dictated  as  a  spelling  test.  One 
pupil  received  a  mark  of  93J  per  cent.  How  many  words 
did  he  misspell  ? 

16.  A  certain  regiment  went  into  battle  with  1000  men. 
Of  these  5%  were  killed,  12%  were  wounded,  3%  were 
taken  prisoners,  and  1%  were  missing.  How  many  re- 
mained available  for  duty? 

17.  What  is  the  duty  at  35  cents  per  square  yard  on  a 
piece  of  cloth  measuring  56  yards,  27  inches  wide  ? 

18.  A  man  bought  a  bill  of  goods  amounting  to  $  374.50, 
with  a  deduction  of  2  %  for  payment  within  10  days.  How 
much  does  he  save  by  paying  the  bill  within  the  10  days  ? 

19.  A  merchant  places  a  bill  of  $  840  in  the  hands  of  a 
collector,  who  collects  75%  of  the  amount.  How  much  does 
the  merchant  receive  if  the  collector  deducts  5%  of  the 
amount  collected,  as  his  commission  ? 

20.  How  many  pounds  of  bread  can  be  made  from  5 
bushels  of  wheat  weighing  60  pounds  per  bushel,  if  the 
wheat  loses  30  per  cent  in  the  process  of  grinding  into  flour, 
and  if  the  bread  weighs  33J  per  cent  more  than  the  weight 
of  the  flour  used  ? 


Measurements, 


*95 


SURFACES. 

276.    Preliminary  Exercises. 

1.  What  is  the  length  in  inches  of  a  row  of  four  enve- 
lopes, each  five  inches  long,  placed  end  to  end  ?  What  is 
the  length  in  feet  and  inches. 


3  inches 

1 

p 

2.  What  is  the  width  in  inches  of  four  such  rows,  each 
envelope  three  inches  wide,  just  touching  each  other  ?  What 
is  the  width  in  feet  ? 

3.  How  many  envelopes  are  there  ?  How  many  square 
inches  are  there  in  each  envelope  ?  How  many  square 
inches  are  covered  by  all  of  them? 

4.  How  many  envelopes  5  inches  by  3  inches  would 
cover  the  top  of  a  table  4  ft.  2  in.  long  and  2  ft.  6  in. 
wide? 

5.  Draw  a  rectangle  to  represent  a  floor  24  feet  long  18 
feet  wide.  Draw  rugs  6  feet  long,  3  feet  wide,  and  see  how 
many  will  be  needed  to  cover  the  floor. 

6.  What  is  the  difference  between  three  square  inches 
and  three  inches  square  ? 

7.  What  is  the  distance  around  a  room  that  is  40  feet 
by  30  feet? 


196  Chapter  Four. 

8.  A  garden  is  12  feet  long  and  9  feet  wide.  How  many 
bunches  of  flowers  will  it  furnish,  if  it  takes  3  square  feet 
to  furnish  one  bunch  ? 

9.  A  room  is  36  feet  long  and  30  feet  wide.  How  many- 
square  yards  in  the  floor  ? 

10.  How  many  yards  is  it  around  a  room  15  feet  long  and 
12  feet  wide  ? 

11.  How  many  square  inches  in  the  surface  of  a  sheet  of 
paper  1  foot  8  inches  long,  11  inches  wide  ? 

12.  How  many  pieces  of  paper  2  inches  square  will  exactly 
cover  a  slate  12  inches  long,  8  inches  wide  ? 

277.   Written  Problems, 

1.  How  many  boards  12  feet  long,  6  inches  wide  will  be 
required  for  a  floor  8  yards  long,  6  yards  wide  ? 

The  floor  is  24  feet  long,  18  feet  wide ;  its  area  in  square  feet  is 
18  x  24.     The  area  of  the  board  in  square  feet  is  12  x  J,  or  6. 

Number  of  boards  =  18  x  24 
6 

Note.  —  Labor  is  frequently  saved  in  examples  involving  multipli- 
cation and  division  by  first  indicating  the  operations  and  then  using 
cancellation. 

2.  How  many  bricks  8  inches  by  4  inches  will  be  needed 
for  a  walk  24  yards  long,  6  feet  wide,  making  no  allowance 
for  waste  ? 

Area  of  top  surface  of  one  brick  =(8x4)  square  inches.  Tne 
length  of  the  walk  in  inches  =  24  x  3  x  12  ;  width  in  inches  =  6  x  12. 
Area  of  walk  in  square  inches  =  24  x  3  x  12  x  6  x  12.  Divide  this 
by  8  x  4,  the  number  of  square  inches  in  the  top  surface  of  a  brick. 

Number  of  bricks  =  24x3x12x6x12 
8x4 

Note.  —  It  will  be  remembered  that  the  divisor  and  the  dividend 
must  be  of  the  same  denomination,  square  inches  in  this  example. 


Measurements.  197 

3.  How  many  paving  tiles  \  foot  square  will  cover  a 
hearth.  6  feet  long,  3  feet  wide  ? 

Make  a  diagram. 

4.  How  many  boards  12  feet  long,  8  inches  wide  will  be 
required  for  a  close  fence  120  yards  long,  6  feet  high  ? 

5.  Find  the  number  of  paving  stones  9  inches  by  3 
inches,  in  a  street  100  rods  long,  10  yards  wide. 

6.  Draw  a  rectangle  2  inches  by  3  inches.  Draw  one 
twice  the  size.  What  are  the  dimensions  of  the  latter? 
What  are  the  dimensions  of  one  four  times  the  size  ? 

A  plot  100  feet  by  100  feet  is  how  many  times  as  large 
as  a  plot  25  feet  by  25  ? 

7.  A  brick  is  8  inches  long,  4  inches  wide,  2  inches  thick. 
How  many  square  inches  are  there  in  the  surface  of  the 
widest  face  ?  In  the  surface  of  one  side  ?  In  the  surface 
of  one  end  ? 

8.  How  many  bricks  laid  on  the  widest  face  will  be 
needed  for  a  walk  28S  inches  long,  96  inches  wide  ? 

9.  How  many  bricks  laid  on  the  side  will  be  needed  for 
a  walk  24  feet  long,  8  feet  wide  ? 

10.  How  many  square  feet  are  there  in  a  roll  of  wall 
paper  24  feet  long,  18  inches  wide  ? 

11.  How  many  rolls  24  feet  long,  1\  feet  wide,  would  be 
required  to  paper  the  ceiling  of  a  room  45  feet  long,  36  feet 
wide,  making  no  allowance  for  matching  or  waste  ? 

12.  The  owner  of  a  piece  of  ground  200  feet  wide,  300 
feet  long,  divides  it  into  lots  25  feet  by  100  feet.  How 
many  lots  are  there  ? 


198  Chapter  Four. 

13.  Make  table  of  square  measure : 

square  inches  (sq.  in. )  =1  square  foot  (sq.  ft.) 
square  feet  =  1  square  yard  (sq.  yd.) 

square  yards  =  1  square  rod  (sq.  rd.) 

160  square  rods  =  1  acre  (A.) 

acres  =  1  square  mile  (sq.  mi.) 

14.  There  are  160  square  rods  in  an  acre.  How  many 
square  yards  are  there  in  an  acre  ? 

15.  Give  the  dimensions,  in  yards,  of  a  field  that  will 
contain  just  an  acre.     Of  one  that  will  contain  two  acres. 

16.  At  $80  per  acre  what  is  the  value  of  a  field  80  rods 
long,  70  rods  wide  ? 

What  will  it  cost  to  fence  the  field  at  20^  per  running 
yard? 

17.  A  man  has  a  lot  100  feet  by  200  feet.  How  many 
square  feet  will  he  have  left  for  a  garden  after  he  builds  a 
house  25  feet  by  60  feet  ? 

18.  One  wall  of  a  room  is  24  feet  long  and  12  feet  high. 
There  is  a  door  in  it  8  feet  high,  4$  feet  wide.  How  many 
square  yards  of  plastering  will  be  needed  to  cover  the  wall  ? 

19.  What  would  be  the  cost  of  painting  1800  feet  of 
fence  6  feet  high  at  15  cents  per  square  yard  ? 

20.  What  is  the  length  of  a  rectangular  field  60  rods 
wide  that  contains  60  acres  ? 

21.  A  farm  is  one  mile  square.  How  many  40-acre  fields 
does  it  contain  ? 

22.  How  many  acres  in  a  field  in  the  shape  of  a  triangle 
whose  base  and  perpendicular  measure  40  rods  each  ? 

23.  How  many  acres  are  there  in  a  triangular  plot  of 
ground  when  the  base  measures  80  yards  and  the  perpen- 
dicular measures  60£  yards  ? 


Measurements. 


199 


3  in.  wide 


VOLUMES. 
278.  Preliminary  Exercises. 

1.  How  many  one-inch  cubes  can  be  placed  on  the  bottom 
of  a  box  4  inches  long,  3  inches  wide  ? 

2.  If  the  box  is  one  inch  high, 
how  many  will  it  hold  ?  If  the 
box  is  2  inches  high  ?  3  inches 
high? 

Note.  —  A  cube  one  inch  long,  one 
inch  wide,  one  inch  high,  contains  a 
cubic  inch. 

3.  How  many  cubic  inches  in  a 
box  3  inches  long,  4  inches  wide, 
1  inch  high  ?     In  a  box  3  inches 
long,  4  inches  wide,  2  inches  high  ? 
4  inches  wide,  4  inches  high  ? 


In  a  box  4  inches  long, 


4.  If  you  had  24  one-inch  cubes,  how  could  you  pile  them 
to  make  a  solid  with  six  rectangular  faces  ? 

5.  If  the  pile  was  2  inches  high,  how  many  cubes  would 
there  be  in  each  tier  ?  How  many  square  inches  would  the 
lower  tier  cover  ? 

6.  How  could  the  24  cubes  be  arranged  to  make  a  pile 
3  inches  high? 

7.  Can  you  give  a  rule  for  finding  the  number  of  cubic 
inches  in  a  box  4  inches  long,  2  inches  high,  3  inches  wide  ? 

8.  How  many  cubic  inches  of  water  would  a  tin  box  hold, 
the  dimensions  of  the  box  being  5  inches  by  3^-  inches  by  4 
inches  ? 

9.  How  many  one-foot  cubes  could  be  placed  in  a  cubical 
box  one  yard  long,  one  yard  wide,  one  yard  high  ? 


200 


Chapter  Four. 


279.  A  solid  has  three 
dimensions :  length, 
breadth,  and  thickness. 

The  volume  or  contents 
of  a  solid,  is  the  space 
it  occupies,  expressed  in 
cubic  inches,  cubic  feet, 
cubic  yards,  etc. 

A  cube  is  a  solid  hav- 
ing six  equal  square 
faces. 

280.  Cubic  Measure. 


1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 


281.   Written  Exercises. 

1.  How  many  cubic  inches  in  a  solid  3  yards  long,  2  feet 
wide,  6  inches  high  ?  How  many  cubic  feet  ?  How  many 
cubic  yards  ? 

To  find  the  volume  in  cubic  inches,  change  3  yards  to  108  inches,  and 
2  feet  to  24  inches. 

Volume  =  (108  x  24  x  6)  cubic  inches. 
Volume  (in  cubic  feet)  =  (9  x  2  x  $)  cubic  feet. 
Volume  (in  cubic  yards)  =  (3  x  §  X  \)  cubic  yards. 

2.  How  many  cubic  feet  of  air  in  a  room  24  feet  long,  18 
feet  wide,  12  feet  high  ? 

3.  Find  the  solid  contents  of  a  piece  of  timber  25  feet 
long,  3  feet  wide,  5  feet  thick.  Is  it  larger  or  smaller  than 
a  piece  4  feet  wide,  4  feet  thick,  and  23  ft.  6  in.  long  ? 


Measurements. 


aoi 


4.  How  many  cubic  yards  of  earth  will  have  to  be  re- 
moved in  digging  a  cellar  18  feet  wide,  55  feet  long,  6  feet 
deep  ?     What  will  be  the  cost  at  60^  a  load  (1  cubic  yard)  ? 

5.  A  brick  is  8  inches  long,  4  inches  wide,  2  inches  thick. 
How  many  bricks  are  there  in  a  pile  60  feet  long,  20  feet 
wide,  5  feet  high  ? 

6.  Find  the  number  of  bricks  in  a  wall  24  feet  wide,  48 
feet  high,  1  foot  thick,  making  no  allowance  for  mortar,  etc. 

7.  How  many  bricks  are  there  to  a  cubic  foot  ? 

8.  Allowing  20  bricks  to  a  cubic  foot  when  laid  in  mortar, 
how  many  bricks  will  be  needed  for  a  wall  24  feet  wide,  50 
feet  high,  20  inches  thick  ? 

9.  What  will  be  the  cost  of  building  a  stone  wall  40  rods 
long,  4  feet  high,  1  yard  thick,  at  $  6.40  per  perch  of  24| 
cubic  feet  ? 

10.  A  cord  of  wood  contains  128  cubic  feet.  If  the  wood 
is  cut  into  4-foot  lengths,  what  should  be  the  other  two 
dimensions  of  a  regular  pile  to  hold  just  a  cord? 

11.  How  many  cords  of  wood  are  there  in  a  pile  24 
feet  long,  4  feet  wide,  12  feet  high  ? 

1  cord  =  128  cubic  feet. 


282.  Cubic  Measure  of 
Capacity. 

231  cu.  in.  =1  gallon 
2150.4  cu.  in.  =  1  bushel 
128  cu.  ft.        =  1  cord 


12.  Find  the  capa- 
city in  bushels  of  a  bin 
1  yd.  long,  2  ft.  4  in. 
wide,  5  ft.  4  in.  high. 


202  Chapter  Four. 

The  capacity  of  a  bin,       3  4        10 

tank,   etc.,  corresponds    to      fifi  x  gg  x  QAj  0         __ 
the  ooZume  of  the  contents  otg^  3         Du*  "  dU  Du*  -***■ 

of  the  bin  or  tank  when  full.  -t »go 

Write  the  dimensions  in  gaa 

inches  as  factors,  with  the  $* 

number  of  cubic  inches  in 
a  bushel  as  a  divisor,  and  cancel. 

The  decimal  point  in  the  divisor  is  moved  one  place  to  the  right, 
and  a  cipher  is  added  to  one  of  the  numbers  above  the  line.  21504  is 
cancelled  by  12,  7,  4,  and  64. 

13.  Find  the  capacity  in  gallons  of  a  tank  1  ft.  9  in.  long, 
1  ft.  3  in.  wide,  1  ft.  10  in.  deep. 

21  x  15  x  22    -,     rLL~i 

— gal.     Cancel. 

14.  How  many  gallons  are  there  in  a  cubic  foot  ? 
Give  the  answer  as  a  mixed  number  ;  as  a  mixed  decimal. 

15.  How  many  cubic  feet  are  there  in  a  bushel  ? 
Give  the  answer  as  a  mixed  number;  as  a  mixed  decimal. 

16.  Give  the  width  of  a  wagon  body  18  inches  high, 
6  feet  long,  that  will  hold,  when  full,  a  cubic  yard. 

17.  A  gallon  contains  231  cu.  in.  Give  the  dimensions 
of  a  tin  box  that  will  hold  exactly  a  gallon. 

18.  A  pile  of  wood  40  feet  long  and  12  feet  wide  con- 
tains 1920  cubic  feet.     How  high  is  it? 

19.  How  much  will  it  cost  to  have  it  cut  if  it  costs  80  cents 
a  cord  ? 

20.  A  pile  of  4-foot  wood  is  16  feet  long  and  6  feet  high. 
Required  the  cost  at  $  5.50  per  cord. 

21.  A  rectangular  tank  is  5  feet  long,  2  feet  wide,  and  2 
feet  deep.     How  many  gallons  of  water  will  it  hold  ? 


Measurements. 


203 


22.  What  is  the  cost  of  digging  a  cellar  21  feet  long,  18 
feet  wide,  and  6  feet  deep,  at  $  .28  a  cubic  yard  ? 

23.  How  much  will  a  block  of  granite  weigh  15  feet  long, 
12  feet  wide,  and  9  feet  thick,  if  9  cubic  feet  weigh  72  lb.? 

SURFACES  OF  RECTANGULAR  SOLIDS. 
283.   Preliminary  Exercises. 

1.  How  many  faces  has  a  cube  ? 

2.  What  is  the  surface  of  each  face  of  an  inch  cube  ? 

3.  How  many  square  inches  are  there  in  all  the  faces  of 
an  inch  cube  ? 

The  accompanying  diagram  shows  the  dimensions  of  a  piece  of 
paper  that  will  exactly  cover  a  square  prism,  whose  base  measures 
4  inches  by  4  inches,  and  whose  height  is  8  inches. 


.9 


.9* 

.9 

4  in. 

J 

00 

4  in. 

4  in. 

.9 
00 

4  in. 

4  in.     J       4  in. 
■!            .9 

ooj                °° 
4  in.     1      4  in. 

.9 

4  h 


4.  How  many  square  inches  are  there  in  the  top  face  of 
the  prism  ?  In  the  bottom  face  ?  In  each  of  the  four  side 
faces  ?  In  the  four  side  faces  ?  In  the  two  ends  ?  In  the 
entire  surface  ? 


204  Chapter  Four. 

284.   Written  Exercises. 

1.  Make  a  diagram  of  a  piece  of  paper  that  when  folded 
will  just  cover  the  six  faces  of  a  brick  8x4x2  inches. 
How  many  square  inches  of  paper  would  be  needed  ? 

2.  The  owner  of  a  piece  of  ground  600  feet  long,  150  feet 
wide,  builds  a  fence  6  feet  high  around  the  plot.  How  many 
square  feet  of  fence  are  there  ? 

The  surface  of  this  fence  may  be  considered  as  the  four  side  faces  of 
a  solid.  The  area  in  square  feet  =  (150  x  6)  +  (600  x  6)  +  (150  x  6) 
-f  (600  x  6).  The  operation  is  shortened  by  adding  150,  600,  150, 
and  600,  and  multiplying  the  sum  by  6.  (1500  x  6)  sq.  ft.  =  9000 
sq.  ft.,  Ans. 

3.  A  room  is  24  feet  long,  18  feet  wide,  12  feet  high. 
Draw,  touching  each  other,  four  rectangles  representing  the 
four  walls.     Write  the  dimensions  of  each  wall. 

What  are  the  dimensions  of  the  large  rectangle  made  up  of  the  four 
smaller  ones?     Give  the  area  in  square  feet.    In  square  yards. 

4.  Show  by  a  diagram  the  shape  of  a  piece  of  paper  that 
when  folded  will  entirely  cover  a  box  12  inches  long,  6 
inches  wide,  4  inches  high.     Write  the  dimensions. 

This  is  called  the  "  development "  of  the  box. 
What  is  the  area  of  the  paper  in  square  inches  ? 

5.  How  many  square  feet  are  there  in  a  fence  10  feet 
high  enclosing  a  lot  250  feet  long,  200  feet  wide  ? 

6.  Make  a  diagram  of  a  room  24  feet  long,  18  feet 
wide,  12  feet  high,  showing  the  surface  that  is  generally 
plastered. 

How  many  square  yards  of  plaster  will  be  needed  for 
the  above  room,  making  no  allowance  for  doors,  windows, 
etc.? 

7.  A  box  is  4  inches  long,  2  inches  wide,  and  2  inches 
deep.  How  many  square  inches  on  its  surface  ?  With  the 
pen,  sketch  a  free-hand  development  of  this  box. 


Measurements.  205 

8.  One  of  the  drawing  models  is  a  square  prism  8  inches 
long  and  4  inches  square.  How  many  square  inches  on  the 
whole  surface  of  the  model  ? 

9.  How  many  square  yards  in  the  walls  of  a  room 
12  feet  wide,  15  feet  long,  and  9  feet  high  ? 

10.  The  floor  of  a  room  is  18J  feet  long,  15^  feet  wide. 
How  many  square  yards  in  the  ceiling  ? 

A  lot  of  land  containing  5250  square  feet  is  125  feet  long. 
How  wide  is  it  ? 


ANGLES,  TRIANGLES,  QUADRILATERALS. 

285.    The  following  may  be  drawn  free-hand,  the  compasses  being 
reserved  for  the  geometrical  problems  in  Chapter  VIII. 

1.  Draw  two  lines  meeting  at  a  point. 
These  lines  make  an  angle. 

2.  Draw  two  lines  that  will  make  four  angles. 

3.  Draw  two  lines  so  as  to  make  two  angles. 
Two  such  angles  are  called  supplementary  anglea 

4.  Make  two  equal  supplementary  angles. 

Equal  supplementary  angles  are  called  right  angles.    A  line  making 
a  right  angle  with  another  line  is  said  to  be  perpendicular  to  it. 

5.  Draw  two  lines  so  as  to  make  one  right  angle. 

Is  the  right  angle  made  by  two  lines,  each  10  feet  long,  any  larger 
than  a  right  angle  made  by  two  lines,  each  1  inch  long  ? 

6.  What  is  the  smallest  number  of  straight  lines  that  will  enclose 
space  ? 

Draw  a  figure  enclosed  by  the  smallest  possible  number 
of  straight  lines.     What  is  its  name  ?     Why  ? 

7.  Make  a  triangle  having  one  right  angle. 


206  Chapter  Four. 

8.  Can  you  draw  a  triangle  having  two  right  angles  ?  Why  ? 
What  name  is  given  to  lines  that  will  not  meet,  no  matter  how  far 
they  are  extended  ? 

9.  An  angle  less  than  a  right  angle  is  called  an  acute  angle. 
Draw  a  triangle  containing  an  acute  angle. 

10.  Can  you  draw  a  triangle  containing  two  acute  angles  ? 
Three  acute  angles  ? 

11.  An  angle  greater  than  a  right  angle  is  called  an  obtuse  angle. 
Draw  a  triangle  containing  an  obtuse  angle. 

12.  Can  you  draw  a  triangle  containing  three  obtuse  angles  ? 
Containing  two  ? 

13.  Draw  a  triangle  with  sides  2  inches,  3  inches,  4  inches, 
respectively. 

A  triangle  having  no  two  sides  equal  is  called  a  scalene  triangle. 

14.  Draw  a  triangle  having  two  equal  sides. 

This  is  called  an  isosceles  triangle.    The  unequal  side  is  called  the 


15.  Draw  an  isosceles  triangle  with  the  base  uppermost. 
With  the  base  on  the  left.     On  the  right. 

16.  Draw  a  triangle  having  three  equal  sides  (an  equilat- 
eral triangle). 

17.  Draw  a  square.      Draw  a  rectangle  4  inches  by  3 
inches. 

How  many  right  angles  in  each  ? 

18.  Draw  a  four-sided  figure  having  its  opposite  sides  par- 
allel, but  containing  no  right  angle  (rhomboid). 

What  kinds  of  angles  does  it  contain  ?    How  many  of  each  ?    Write 
name  in  each  angle. 

19.  Draw  a  four-sided  figure,  having  all  its  sides  equal, 
but  containing  no  right  angle  (rhombus). 


Measurements,  207 

20.  Draw  a  quadrilateral  (four-sided  figure)  having  only 
two  parallel  sides  (trapezoid). 

21.  Draw  a  quadrilateral  having  no  parallel  sides  (trape- 
zium). 

22.  Draw  a  rhombus,  each  side  2  inches.  A  square,  each 
side  2  inches. 

What  is  the  difference  between  them  ?    Which  is  larger  ? 

23.  A  parallelogram  is  a  quadrilateral  that  has  its  opposite 
sides  parallel. 

Name  the  parallelograms  that  have  four  equal  sides  (equilateral) , 
Those  that  have  four  equal  angles  (equiangular). 

24.  The  height  of  a  parallelogram  is  called  its  altitude. 
Draw  a  rectangle,  base  3^-  inches,  altitude  2-^-  inches.  Draw 
a  rhomboid,  base  3^-  inches,  altitude  2  J  inches.  Draw  several 
rhomboids  of  the  above  dimensions,  all  differing  in  shape. 

25.  Cut  out  of  paper  a  rectangle,  base  3  inches,  altitude 
2  inches.  Cut  out  a  rhomboid,  base  3  inches,  altitude  2 
inches.  Place  one  upon  the  other,  and  see  how  their  areas 
compare. 

26.  Can  you  calculate  the  number  of  square  inches  in  a 
rhomboid  whose  base  is  3  inches  and  altitude  2  inches  ? 

27.  Draw  a  rectangle,  base  4  inches,  altitude  3  inches. 
Divide  by  a  diagonal  into  two  triangles.  Mark  in  each 
triangle  its  area. 

28.  Draw  a  right-angled  triangle,  base  4  inches,  perpen- 
dicular (altitude)  3  inches.     Calculate  its  area. 

29.  Draw  a  rectangle,  base  4  inches,  altitude  3  inches. 
From  the  middle  point  of  the  upper  base  draw  lines  to  the 
extremities  of  the  lower  base,  making  three  triangles.  Mark 
in  each  triangle  its  area. 

30.  Draw  an  isosceles  triangle,  base  4  inches,  altitude  3 
inches,  and  calculate  its  area. 


208  Chapter  Four. 

286.  Areas  of  Triangles  and  Quadrilaterals. 
Find  the  areas  of  the  following : 

1.  A  right-angled  triangle  whose  sides  meas- 
ure 15,  20,  and  25  inches  respectively. 

Note.  —  Area  of  triangle  =  £  product  of  base  by  altitude  (perpen- 
dicular). 

2.  A  right-angled  triangle  whose  base  measures  64  yards, 
perpendicular  48  yards. 

3.  A  triangle  whose  base  measures  18  rods,  altitude  13 
rods. 

4.  A  square  whose  side  measures  35  feet. 

Area  of  parallelogram  =  base  x  altitude. 

97  ft. 


5.  A  rectangle  42  yards  by  37  yards.  /  j^ 

/  Is 

6.  A  rhombus  whose  base  is  97  feet,      Z I 

altitude  63  feet. 

Show  that  the  area  of  this  parallelogram  is  equal  to  that  of  a 
rectangle  97  feet  by  63  feet. 

7.  A  rhomboid,  base  33  meters,  altitude  28  meters. 

8.  A  trapezoid  whose  paral- 
lel sides  measure  10  and  16  feet, 
respectively,  the   perpendicular     j 
distance  between  them  being  6 
feet.  E 


10  ft. 


10  ft. 


U    c 


Draw  this  trapezoid  on  a  scale  of  \  inch  to  the  foot,  and  measure 
AB,  which  divides  the  rectangle  EFOH  into  two  equal  parts.  AB  = 
\(FG  +  ED).  Cut  off  the  triangle  BCD  and  add  it  to  the  upper  half 
of  the  trapezoid,  so  that  CD  will  be  a  continuation  of  FO.  The  rec- 
tangle thus  formed  should  measure  13  feet  by  6  feet. 


Review. 


209 


9.   A  trapezoid  as  shown 
in  the  accompanying  diagram. 

Draw  to  a  scale ;  cut  off  a  triangle 
from  A  to  the  centre  of  CD,  also 
one  from  B  to  the  centre  of  EF\ 
and  place  these  triangles  above  AB,  so  as  to  make  a  rectangle, 
£(10  +  16)  feet  long  and  6  feet  wide. 

10.  A  trapezium,  one  of 
whose  diagonals  measures  42 
yards,  the  perpendiculars  to  the 
opposite  corners  measuring  18 
yards  and  16  yards,  respec- 
tively. 

Area  in  square  yards  =  (42  x  \  of  18)  +  (42  x  \  of  16)  =  42  x  \  of 
(18  +  16). 

SPECIAL  DRILLS.— REVIEW. 

287.   Oral  Exercises. 

1.    463  +  157  =  463  +  100  +  60  +  7  = 
In  giving  the  solution  at  sight,  the  pupil  says  (or  thinks)  663,  613, 


620. 


2.  256  +  184     4.  185  +  546      6.  167  +  734 

3.  419  +  342     5.  668  +  193      7.  476  +  155 

8.  4170  +  470  =  4170  +  400  +  70 

Use  no  unnecessary  words :  4570,  4640. 

9.  1260  +  850         11.   3450  +  390  13.   5620+590 

10.   2140  +  680         12.   4370  +  280  14.   6380  +  660 

15.      400  —  163  =  400  -  100  -  60  -  3  = 
Say  only  300,  240,  237. 


2io  Chapter  Four. 

16.  501-375  18.    650-488  20.    361-149 

17.  275-137  19.   540-384  21.   455-358 

22.  7310  —  6850  =  7310  -  6800  -  50  = 

510,  460. 

23.  8610-7680       25.    4960-4380       27.    6450-5760 

24.  5000-4670       26.    2770-1890       28.    7320-6560 

29.  24  X  66%  =  f  of  24  hundred. 

30.  48  x  16}  33.    24  x  62£  36.  28  x  75 

31.  32  x  37i-  34.    36  x  66|  37.  40  x  87£ 

32.  49  x  25  35.    39  x  33^  38.  88  x  12£ 

39.  533£  -*-  66%  =  5£  hundred  --  f  hundred  =  16  -*-  2. 

40.  337^-^-371         42.   687± -- 62|  44.   437£--87£ 

41.  733J-r-33£         43.    933|-f-66f  45.    212£-12| 

288.   Oral  Problems. 

1.  How  many  ounces  in  11^-  pounds  ? 

2.  258  yards  equal  how  many  feet  ? 

3.  A  dealer  bought  652  tons  of  coal  and  sold  476  tons. 
How  much  had  he  left  ? 

4.  Sold  my  wheat  for  $  347  and  my  oats  for  $  154.     How 
much  did  I  receive  for  both  ? 

5.  40|  yards  of  ribbon  are  cut  into  7  pieces.     Find  the 
length  of  each  piece. 

6.  How  many  square  yards  in  a  floor  5£  yards  long  and 
6£  yards  wide  ? 

7.  What  will  be  the  cost  of  14  pounds  of  lard  at  14^  per 
pound? 

8.  At  1\$  each,  how  many  lead  pencils  can  I  buy  for  27^  ? 


Review.  211 

9.  What  part  of  a  196-pound  barrel  of  flour  is  contained 
in  a  49-pound  bag  ? 

10.  At  45^  per  yard,  bow  much  lace  can  be  bought  for 
$1.35? 

11.  A  woman  has  saved  $  833.     How  much  more  must 
she  save  to  have  $1000? 

12.  What  will  be  the  cost  of  16  pounds  of  sugar  at  4f^ 
per  pound  ? 

13.  Spent  $  2.56  for  dry  goods  and  $  1.84  for  groceries. 
How  much  did  I  spend  for  both  ? 

14.  Find  the  cost  of  3  lb.  10  oz.  butter  at  32^  per  pound. 

15.  At  $.375  per  yard  how  much  ribbon  can  be  bought  for 

$.75? 

16.  If  it  takes  1-J  yards  of  cloth  to  make  a  jacket,  how  many 
can  be  made  from  a  piece  of  cloth  containing  30  yards  ? 

17.  A  boy  paid  35^  for  the  use  of  a  boat  for  3J  hours. 
What  was  the  price  per  hour  ? 

18.  If  13  pounds  of  raisins  cost  $1.69,  what  is  the  cost  of 
1  pound  ? 

APPROXIMATIONS. 
289.   Give  an  estimate  of  the  answer : 

1.  If  3  T.  1988  lb.  coal  cost  $19.97,  what  will  be  the 
cost  of  8  T.  1  lb.? 

Nearly  4  tons  cost  nearly  $20. 

2.  At  $  500  per  year,  what  will  be  the  rent  of  a  house  for 

1  yr.  11  mo.  29  da.? 

Nearly  2  years. 

3.  Find  the  cost  of  5  barrels  sugar,  averaging  299  pounds 
each,  at  4ff ^  per  pound. 

4.  What  is  the  interest  on  $199.86  at  6%,  for  5  ma 
28  da.? 


212  Chapter  Four. 

5.  If  11  men  and  2  boys  can  finish  a  piece  of  work  in 
23£  days,  how  long  will  it  take  23  men  and  5  boys  ? 

6.  What  decimal  of  639  acres  is  321  acres  ? 

7.  What  will  be  the  cost  of  20,060  bricks  at  $  4.90  per  M  ? 

8.  A  farmer  sells  5484  pounds  rye  at  87^  per  bushel  of 
56  pounds.     How  much  does  he  receive  ? 

9.  If  19  lb.  15  oz.  of  tea  cost  $  7.95,  what  will  be  the  cost 
of  21  lb.  1  oz.? 

10.  Paid  freight  on  1987  pounds  at  70^  per  cwt.  How 
much  did  I  pay  ? 

11.  If  there  are  about  1\  gallons  to  a  cubic  foot,  estimate 
the  number  of  gallons  in  a  tank  5  feet  long,  3  feet  wide, 
4  feet  high. 

12.  If  there  are  about  \\  cubic  feet  in  a  bushel,  estimate 
the  contents  in  bushels  of  a  bin  5  ft.  x  3  ft.  x  4  ft. 

13.  Give  the  dimensions  of  a  tank  of  150  gallons'  capacity. 

14.  Give  the  dimensions  of  a  bin  that  will  hold  100 
bushels. 

15.  At  20  bricks  laid  in  mortar  to  the  cubic  foot,  give  the 
length  and  the  height  of  a  wall  1  foot  thick  that  can  be  built 
with  a  thousand  bricks. 

16.  At  $  1  a  load  (1  cubic  yard),  give  the  dimensions  of 
an  excavation  that  can  be  made  for  $  100. 

17.  A  cubic  foot  of  water  (about  1\  gallons),  weighs  62^- 
pounds.     About  what  does  a  gallon  weigh  ?     A  pint  ? 

18.  If  iron  is  about  1\  times  as  heavy  as  water,  about 
what  does  a  cubic  foot  of  iron  weigh  ? 

19.  About  what  is  49f%  of  f  801? 

20.  About  what  will  be  the  interest  at  6  per  cent  on  %  100 
for  3  yr.  11  mo.  29  da.? 


Review.  213 


FUNDAMENTAL  PROCESSES. 

290.   1.   The  sum  of  two  numbers  is  278.      One  of  the 

numbers  is  89.     What  is  the  other  ? 
89  +  ?  =  278 

2.  The  minuend  is  583,  the  remainder  is  249.     What  is 
the  subtrahend  ?  583  -  ?  =  249 

3.  The  subtrahend  is  56,  the  minuend  is  214.     Find  the 
remainder. 

4.  The  difference  between  two  numbers  is  84,  the  smaller 
is  129.     What  is  the  larger  number  ? 

5.  The  subtrahend  is  176,  the  remainder  is  92.     Find 
the  minuend. 

6.  The  multiplier  is  98,  the  multiplicand  is  809.     Find 
the  product. 

7.  The  product  is  9045,  the  multiplier  is  45.     What  is 
the  multiplicand? 

8.  The  product  of  two  factors  is  1767.      One  of  the 
factors  is  93.     Find  the  other  factor. 

9.  The  multiplicand  is  84,  the  product  is  2100.     What 
is  the  multiplier  ? 

10.  The  dividend  is  10,000,  the  divisor  is  275.     Find  the 
remainder. 

11.  The  quotient  is  32,  the  remainder  is  21,  the  divisor 
is  40.     What  is  the  dividend  ? 

40)    ? 

12.  The  dividend  is  4263,  the  quotient  is  203.     Find  the 
divisor.  4263  _  2Q3 

? 

13.  The  dividend  is  267,  the  quotient  is  13,  the  remainder 
is  7.    What  is  the  divisor  ? 

267     ,  o* 


H 

Chapter 

Four. 

RATIO. 
291.   Sight  Exercises. 

-     87x25 
*        75 

0  63x19 
3*       21 

5.    gWil 

7.   1x55 

«,    74x24 
2*       37 

A  96  x  27 
4'        32 

,   gx42 

8.   ^x32 

292.  Written  Exercises. 

Indicate  operations,  and  cancel  where  possible.  Terms  compared 
*hould  be  of  the  same  denomination. 

1.  If  90  tons  of  coal  cost  $472.50,  what  will  be  the  cost 
of  132  tons  ?  $472.50x132 

90 

2.  If  3  lb.  4  oz.  tea  cost  $  1.95,  what  will  12  oz.  cost  ? 

The  ratio  is  12  oz.  to  52  oz. 

3.  A  party  of  men  can  build  16  rd.  2  ft.  of  wall  in  20 
days.     How  long  will  it  take  them  to  build  4  yd.  6  in.  ? 

Change  to  inches. 

4.  What  will  be  the  cost  of  3  bu.  2  pk.  7  qt.  of  oats  if  7 
bu.  1  qt.  cost  $4.50? 

5.  By  travelling  at  the  rate  of  20  miles  a  day,  a  person 
■can  complete  a  journey  in  18  days.  At  what  rate  must  he 
travel  to  finish  it  in  15  days  ? 

6.  How  many  rolls  of  merino,  each  containing  75  yards, 
'worth  $  .45  per  yard,  will  it  take  to  pay  for  180  yards  of 
alpaca  at  $  .30  per  yard  ? 

7.  A  merchant  sold  20  hogsheads  of  oil,  each  containing 
■63  gallons,  at  $  1.75  per  gallon,  and  invested  the  proceeds  in 
table  sauce  in  cases  of  12  bottles  each,  worth  $.31 J  per 
bottle.     How  many  cases  did  he  buy  ? 


Review.  215 

8.  No  allowance  being  made  for  mortar,  how  many- 
bricks  will  be  required  to  build  a  wall  50  feet  long,  4  feet 
high,  and  1  foot  3  inches  thick,  each  brick  being  8  inches 
long,  4  inches  wide,  and  2\  inches  thicK '/ 

9.  If  .1875  of  a  vessel  cost  % 273.12J,  what  is  the  value 
of  -^  of  it  at  the  same  rate  ? 

10.  What  is  the  cost  of  60.51  tons  of  coal,  when  .9  of  a 
ton  costs  §6.66? 

REVIEW  OF  FRACTIONS. 
293.   Add  across : 

If  the  pupils  work  from  their  books  the  following  examples  in 
addition  and  subtraction,  they  should  be  permitted  to  write  only  the 
answers.  The  teacher  should  announce  the  number  o*  an  example,, 
not  taking  them  in  order,  then  the  number  of  the  next  «o  be  worked,, 
without  giving  time  for  the  writing  of  unnecessary  figures. 

1.  13i-4-16|  +  8f  6.  59f  +  3£  +  4f 

2.  4i  +  5f  +  27f  7.  7f-f-18f  +  40i 

3.  19i  +  3f  +  35£  8.  35f  +  5H  +  8^ 

4.  81  +  9^  +  14!  9.  3J  +  9J  +  25^ 

5.  23f  +  5J  +  32^  10.  66J  +  8f  +  14i 


294.    Subtract  across : 

11.   25\   -18^ 

16. 

68|   -6{i 

12.    63|   -49f 

17. 

100£  -  62J 

13.   70^-15£ 

18. 

56{   -37£ 

14.   92f   -24J 

19. 

83|.   _43| 

15.   33J   -15^ 

20. 

42J    -16} 

2i 6  Chapter  Four. 


Multiply : 

When  the  fractions  are  small  and  the  fraction  in  the  multiplicand 
has  1  for  its  numerator,  business  men  do  not  change  the  mixed  num- 
bers to  improper  fractions. 

In  multiplying  38f  by  11,  the  product  of  f  by  11  is  mentally  reduced 
to  8£,  and  \  written ;  11  eights  (88),  and  8  (96),  6  being  written  ;  etc. 
\  of  38|  is  4  (written)  with  6f  remainder.  This  is  reduced  to  ^  men- 
tally, and  its  },  or  ||,  written.* 


37}  x  3} 

12f  X  5} 

38 

fxlli 

1121- 
18| 

63f 

426} 

131}  Ans. 

68   Ans. 

431 

h    Ans- 

21. 

48}  x  4} 

24.   18}  x  5} 

27. 

45}  x  2i 

22. 

64} . :  10} 

25.    13}  x  7} 

28. 

50}  x  10} 

23. 

29f  x  6} 

26.   9fx8} 

296 

1.   Divide: 

29. 

13)2051 

The  pupil  should  endeavor  to  work  the  following  by  short  division  : 
into  20,  once ;  into  75,  5  times,  remainder  10$  or  -^  •  ^  of  -^  =  f£. 

Ans.  16ft. 

30. 

14)186} 

37.    21)450} 

44. 

25)568} 

31. 

15)250} 

38.    31)970} 

45. 

32)965} 

32. 

16)198} 

39.    24)553$ 

46. 

36)722£ 

33. 

17)190£ 

40.    23)466f 

47. 

16)366} 

34. 

18)200} 

41.    26)290£ 

48. 

17)208} 

35. 

19)381} 

42.    27)545 J 

49. 

21)640} 

36. 

22)264} 

43.   33)9994 

50. 

22)888} 

Review.  217 


REVIEW  OF  DECIMALS. 

297.  Sight  Exercises. 

Give  products : 

1.  360  x. 25 

8. 

840  x  .075 

15. 

400  x  .04 

2.  560  x. 125 

9. 

960  x  .005 

16. 

165  x  .06| 

3.  240  x. 375 

10. 

1200  x  .001 

17. 

176  x  .06^ 

4.  400  x  .625 

11. 

1500  x  .002 

18. 

3300  x  .00£ 

5.  480  x. 75 

12. 

96  x  .3£ 

19. 

880  x  .12£ 

6.  320  x. 875 

13. 

840  x  .02£ 

20. 

105  x  .8 

7.  720  x. 025 

14. 

1500  x  .06 

21. 

210  x  .10 

298.  Give  quotients : 

1.  240  -h.5 

8. 

37  -*-  .05 

15. 

76  -*-  .04 

2.  360 -i- .75 

9. 

48  -T-  .005 

16. 

88  -*-  .00£ 

3.  45  -.125 

10. 

72  -  .025 

17. 

65  -*•  .12^ 

4.  23  -h.25 

11. 

92  -  .002 

18. 

84-^.8 

5.  360 -.375 

12. 

93  -  .03J 

19. 

11  +  .06J 

6.  100 -.625 

13. 

54  -v-  .02£ 

20. 

42 -.6£ 

7.  154 -.875 

14. 

132  -r-  .06 

21. 

93-^.5 

299.   Written  Exercises. 

1.  Find  the  value  of  (6.125  +  8.75  -  9.1235)  -r-  .0125. 

2.  Find  the  value  of  (1708.4592  -  .00024)  x  .003. 

4.  Multiply   24.234  by   .346,   and  write    the  result  in 
words. 

5.  Divide  96  ten-thousandths  by  384  hundred-millionths. 


ai8  Chapter  Four. 

6.  Why  does  the  value  of  a  decimal  remain  unchanged 
when  ciphers  are  annexed  ? 

7.  Write  :  four  hundred  seven  thousandths. 

8.  Write :  six  hundred  four  millionths. 

9.  Write  in  words  405.0067542. 

10.  Reduce  to  common  fractions  in  lowest  terms: 

.004;  .0125;  56.37$. 

11.  16f  x  .045  =  ?    .324  x  .33J  =  ?    3.406  x  1.00  =  ? 

12.  .805  -^  .35  =  ?     80.5  -f-  350  =  ?     Divide  twenty-five 
thousandths  by  16  millionths. 

13.  Write  in  words : 

.0105;  000125;  1.001105;  11.4141;  .000008. 

14.  Reduce  to  common  fractions :  .95 ;  .526. 

15.  From  one  thousand  and  (decimal)  five  thousandths 
take  eight  hundred  and  (decimal)  eight  hundredths. 

16.  Divide  eight  hundredths  by  four  thousandths,  and 
multiply  the  quotient  by  six  ten-thousandths. 

17.  Find  the  product   of   the  following  factors:   .064, 
,0032, 15,625,  and  31.25. 

300.   Oral  Eeview  Problems. 

1.  At  20^  per  quart,  what  will  be  the  cost  of  2  gal.  3  qt. 
1  pt.  of  maple  syrup  ? 

2.  Find  the  cost  of  4  T.  400  lb.  of  coal  at  $  5  per  ton. 

3.  A  man  puts  4  lb.  8  oz.  of  tea  into  9-ounce  packages. 
How  many  packages  does  he  make  ? 

4.  4  pk.  3  qt.  of  apples  are  given  to  some  children.     If 
each  child's  share  is  5  quarts,  how  many  children  are  there  ? 

5.  If  it  takes  3  hr.  20  min.  to  hoe  a  row  of  corn,  how 
many  rows  can  a  man  do  in  2  days  of  10  hours  each  ? 


Review.  219 

6.  How  many  dozen  eggs  at  25^  a  dozen  must  be  given 
for  100  pounds  of  sugar  at  5fi  a  pound  ? 

7.  Which  would  you  rather  have,  -J  of  a  dollar  or  75^  ? 
Why? 

8.  What  will  a  gallon  of  molasses  cost  if  a  gill  costs  2\f  ? 

1  gill  =  £  pint 

9.  Give  the  names  to  the  results  in  the  four  simplest 
processes  in  arithmetic. 

10.  $15  per  week  is  how  much  per  day  ? 

11.  I  of  72  is  f  of  what  number? 

12.  How  many  cubic  feet  in  -|  of  a  cubic  yard  ? 

13.  Which  is  the  larger  and  how  much  larger,  |  of  130  or 
f  of  119? 

14.  Which  is  the  larger  and  how  much,  ^  or  f  ? 

15.  How  many  cubic  feet  in  a  wall  30  feet  long,  4  feet 
high,  and  2  feet  thick  ? 

.16.    Iff  of  a  barrel  of  flour  cost  $2.13,  what  cost  1^ 
barrels  ? 

•  17.    The   difference  between  144   and  24  is  how  many 
times  15  ? 

•  18.   John  walked  12f  miles,  and  Henry  lOf  miles.     How 
much  farther  did  John  walk  than  Henry  ? 

19.  At  41^  a  pint,  what  will  5  qt.  1  pt.  of  milk  cost  ? 

20.  After  spending  f  of  his  money,  James  has  $  150  left. 
What  amount  did  he  have  at  first  ? 

21.  How  many  gallons  in  462  cubic  inches  ? 

22.  If  a  boy  eats  f  of  a  loaf  of  bread,  how  many  boys 
will  be  required  to  eat  10  loaves  ? 

23.  5  yd.  cloth  cost  90^ ;  find  the  cost  of  f  yd. 

24.  If  |  yd.   of  cloth  costs  10^,  how  many  yards  can. 
b©  bought  for  80^  ? 


220  Chapter  Four. 

25.  A  step  is  3  feet.     2  steps  are  what  part  of  a  rod  ? 

26.  19+3  +  17  +  6  +  15  +  4=? 

27.  John  had  85^.  He  bought  strawberries  for  22^-; 
1  pound  coffee  for  30^ ;  3  sheets  paper  at  1^  a  sheet.  What 
remained  ? 

28.  Three-fourths  of  a  mince  pie  is  worth  18^,  and  James 
eats  ^  of  a  pie.     What  is  the  value  of  what  he  eats  ? 

29.  If  I  have  1  pk.  2  qt.  1  pt.  of  meal,  how  many  more 
quarts  must  there  be  to  make  1  bushel  ? 

30.  Charles  caught  12  fish,  worth  41  ^  each,  in  four 
hours.  His  time  was  worth  12^  an  hour.  Gain  or  loss,  and 
how  much  ? 

31.  How  many  times  would  a  dish  holding  f  of  a  pint 
have  to  be  filled  to  measure  9  quarts  ? 

32.  If  5  chairs  cost  $  80,  what  will  12  chairs  cost  ? 

33.  How  many  hours  from  4  a.m.  to  8  p.m.  ? 

34.  Eeduce  ff  to  lowest  terms. 

35.  Add  -J-  to  |,  and  take  the  sum  from  5. 

301.   "Written  Eeview  Problems. 

1.  What  part  of  6  hr.  54  min.  are  3  hr.  15  min.  ? 

2.  If  a  man  walks  at  the  rate  of  3  mi.  96  rd.  per  hour, 
how  far  will  he  walk  in  3  hr.  20  min.  ? 

3.  What  is  one-ninth  of  28  bu.  3  pk.  7  qt.  ? 

4.  Three  men  buy  a  house  for  $  1200.  A  furnishes 
$600;  B,  $400;  C,  $200.  They  sell  the  house  for  $1500. 
How  much  money  should  each  receive  ? 

5.  If  5  T.  1000  lb.  of  coal  cost  $30.25,  how  much  will 
be  paid  for  7  T.  320  lb.  ? 

6.  At  25^  per  hour,  how  much  should  a  man  receive 
that  works  8  hours  and  36  minutes  ? 


Review.  221 

7.  If  2  lb.  4  oz.  of  tea  cost  $1.35,  what  will  be  the  cost 
of  11  lb.  12  oz.  ? 

8.  How  many  square  inches  in  a  paving  tile  6  inches 
square  ?  How  many  square  inches  in  a  rectangle  4  feet  by 
3  feet  ?  How  many  paving  tiles  6  inches  by  6  inches  would 
cover  a  surface  4  feet  by  3  feet  ? 

9.  A  man  buys  a  house  and  lot  for  $3000.  He  pays  -f 
of  the  amount  in  cash  and  the  remainder  after  1  year,  4 
months,  with  5%  interest.  Find  the  amount  of  the  second 
payment. 

10.  Find  four-ninths  of  28  bu.  3  pk.  7  qt. 

11.  (fof  D  +  (fof  0-Cftcf  2)  =  ? 

10  tiia    AQf4f__* 

I  of  15      1J  x  11      ' 

13.  AU.8f+f+f+.A,tt 

14.  Find  the  value  of  728  - 1  -  \  -  £  -  \. 

15.  1}.X  <&.-**)  X& 

16.  Eeduce  ^  of  f  of  £  of  ^^  to  a  decimal. 

17.  A  person  owning  ^  of  a  factory  sells  75  per  cent  of 
his  share  for  $  1710.  What  is  the  value  of  the  whole  fac- 
tory? 

18.  Find  f  of  2  da.  5  hr.  40  min. 

19.  If  a  piece  of  cloth  is  20  yards  long  and  f  yard  broad, 
how  broad  is  another  piece  which  is  12  yards  long  and  con- 
tains as  many  square  yards  as  the  first  ? 

so.    Simplify|l±l|x^. 

*  21.  If  7  men  can  do  a  piece  of  work  in  10 J  days,  how  long 
will  it  take  8  men  and  5  boys  to  do  the  same  work,  each  boy 
doing  one-half  as  much  as  a  man  ? 


222  Chapter  Four. 

22.  A  farmer  drew  to  market  three  loads  of  wheat, 
weighing  respectively  2873  pounds,  3027  pounds,  and  2911 
pounds.  At  93^  per  bushel  (60  pounds),  how  much  did  he 
receive  for  the  three  loads  ? 

23.  How  many  acres  of  land  are  there  in  a  rectangular 
farm  J  of  a  mile  long  and  f  of  a  mile  wide?  (1  square 
mile  =  640  acres.) 

24.  Eeduce    ^""^   *  to  a  simple  fraction. 

25.  The  sum  of  two  numbers  is  15f,  and  one  of  them  is 
9^5".     Find  the  other  number. 

26.  If  3  be  added  to  both  terms  of  the  fraction  -f,  will 
the  value  be  increased  or  diminished,  and  how  much? 

27.  Make  and  solve  a  problem  to  illustrate  reduction 
descending;  one  to  illustrate  reduction  ascending. 

28.  How  is  the  value  of  a  fraction  changed  by  increasing 
its  denominator?    Why? 

29.  Add  %  hours,  20|  minutes,  and  49.2  seconds.  Ex- 
press the  answer  in  minutes  and  seconds. 

30.  What  fractional  part  of  31^  is  12|? 

31.  In  a  hotel  the  weekly  wages  of  the  clerk  are  $  15,  of 
the  cook  $  7.50,  of  the  porter  $  9,  of  the  waiter  $  3,  of  the 
hostler  $6,  and  of  the  errand  boy  $4.  Find  the  average 
wages  paid. 

32.  A  man  was  born  May  24,  1832.  What  is  his  age 
to-day? 

33.  A  grocer's  bill  for  $  84.36  is  paid  8  months  15  days 
after  it  becomes  due,  with  interest  at  5%.     How  much  is 


34.    Find  the  cost  of  7  lb.  11  oz.  of  cheese  at  13^  per 
pound. 


Review.  223 

35.  Find  the  cost  of  digging  a  cellar  30  feet  long,  15  feet 
wide,  and  5  feet  deep,  at  30^  per  cubic  yard. 

36.  John  Smith  bought  of  Clark  and  Jones, 

4  lb.  13  oz.  beefsteak  @  21^  per  lb. 
12  lb.  of  bacon  @  12j£ 

Make  a  properly  receipted  bill  of  the  above,  dated  at  the 
time  and  place  of  this  lesson. 

37.  Find  the  cost  o   2315  pounds  of  coal  at  $  5.75  per  ton. 

38.  Write  1249  in  1  oman  notation. 

39.  Given  the  dividand  807  and  the  quotient  34 J,  find 
the  divisor. 

40.  What  will  it  cost  to  fill  a  jug,  which  contains  2310 
cubic  inches,  with  vinegar  at  7  cents  a  quart  ? 

(1  gal  =  231  cu.  in.) 

41.  Mrs.  C.  B.  Jones  bought  of  Cole,  Steele,  &  Co.,  of 
Indianapolis,  as  follows:  Nov.  12,  1904,  23  yards  of  muslin 
@  16|^;  45  yards  of  sheeting  @  12^  ;  Dec.  7,  12  yards 
of  silk  @  $1,621^;  8  handkerchiefs  @  45^;  2  pairs  kid 
gloves  @  $  1.371 ;  6  neckties  @  75^.  Make  out  and  receipt 
the  above  bill. 

42.  If  a  boy  bought  f  of  a  bushel  of  nuts  for  $2.00,  and 
sold  them  for  12^  a  quart,  what  was  his  gain  ? 

43.  Reduce  -^  of  an  inch  to  the  fraction  of  a  rod. 

44.  Eeduce  35  quarts  to  the  fraction  of  a  barrel  (31 J 
gal.). 

3450  cubic  feet  to  cubic  yards. 

45.  Put  the  following  in  the  proper  form  of  a  bill,  find 
the  amount  of  the  bill,  and  receipt  it : 

David  Wilson  bought  of  Harry  Lloyd,  June  10,  1904, 
7  pounds  of  oatmeal  at  6^  a  pound ;  10  pounds  of  sugar  at 
7$j  a  pound ;  14  pounds  of  ham  at  13£^  a  pound ;  3  brooms 
at  $  2.25  a  dozen. 


224  Chapter  Four. 

46.  A  family  uses  2  quarts  of  milk  a  day.  At  24/  a 
gallon,  what  does  the  milk  cost  for  May  and  June  ? 

47.  From  March  3d  to  Sept.  19th  is  how  many  days? 
Do  you  include  one  of  the  days  mentioned,  or  both  of  them, 
or  neither  of  them  ? 

48.  How  many  minutes  from  8.10  a.m.  to  9.25  p.m. 

49.  Subtract  40  rd.  3  yd.  2  ft.  from  81  rd.  1  yd.,  and 
multiply  the  remainder  by  10.  Work  by  compound  sub- 
traction and  multiplication,  and  get  an  answer  that  contains 
no  fraction. 

50.  Draw  and  divide  a  figure  so  as  to  show  how  many 
square  feet  in  a  rectangle  that  is  5  feet  long  and  3  feet 
wide.  Draw  and  divide  a  figure  so  as  to  show  how  many 
square  inches  in  a  surface  that  is  4  inches  square.  These 
drawings  are  to  be  free-hand,  and  made  with  your  pen. 

51.  Keduce  7  months  and  15  days  to  the  decimal  of  a 
year  (360  days). 

52.  Eeduce  .32175  of  1  ton  to  whole  numbers  of  lower 
denominations. 

53.  If  the  perimeter  of  a  square  is  10  rods,  what  is  the 
area? 

Find  the  area  of  a  field,  whose  parallel  sides  measure 
20  and  30  rods,  respectively,  the  perpendicular  distance 
between  them  being  15  rods. 

54.  Bought  5  bushels  of  berries  for  $  5  and  sold  them  at 
«8>  .20  a  quart.     How  much  did  I  gain  ? 

65.  From  a  tract  of  land  15  rods  square  I  sold  65  square 
rods.    What  was  the  value  of  the  remainder  at  $  20  an  acre  ? 

66.  What  is  the  cost  of  fencing  a  lot  9  rods  square  at 
$ .12  a  foot? 


Review.  225 

57.  How  many  square  yards  are  there  in  the  walls  of  a 
rocm  20  feet  long,  18  feet  wide,  and  9  feet  high  ? 

5S.  What  must  I  pay  for  the  laying  of  a  sidewalk  6  rods 
long  and  5  feet  wide  at  $  .45  a  square  yard  ? 

59.  How  much  will  it  cost  to  plaster  a  room  18  feet  long, 
15  feet  wide,  and  9  feet  high,  at  $  .17  a  square  yard,  deduct- 
ing 108  square  feet  for  doors  and  windows  ? 

60.  Mr.  Thompson  has  a  field,  around  which  he  wishes  to 
build  a  tight  board  fence.  The  field  is  50  rods  long  and  45 
rods  wide.  The  fence  is  to  be  4J  feet  high.  At  3JP  a  square 
foot,  what  will  be  the  cost  of  the  fence  ? 

61. .  A  man  having  $  100  went  to  market.  He  sold  10 
bushels  of  potatoes  at  80^  per  bushel,  2  tons  of  hay  at 
$  15  per  ton,  and  25  bushels  of  oats  at  45^  per  bushel.  He 
bought  15  barrels  of  flour  at  $  4.50  per  barrel,  and  12  yards 
of  broadcloth  at  $  4.75  per  yard.  How  much  money  did  he 
have  left? 

62.  Cost  of  a  pile  of  wood  10  feet  long,  4  feet  wide,  and 
^  feet  high,  at  $  7.50  a  cord  ? 

I  wish  to  pile  60  cords  of  wood  in  such  a  manner  that  it 
will  be  4  feet  wide  and  6  feet  high.     How  long  must  it  be  ? 

63.  Find  the  interest  of  1 263.75  for  1  year,  3  months, 
20  days,  at  6%. 

64.  At  $  17.625  a  ton,  how  many  tons  of  hay  can  be  pur- 
chased for  $  95  ? 

65.  Mr.  Ames  owns  \\  of  an  acre  of  land.  Mr.  Jones 
owns  -|  as  much,  which  is  \  of  what  Mr.  Brown  owns. 
What  part  of  an  acre  does  Mr.  Brown  own? 

66.  Four  men  built  a  barn.  A  worked  2  days ;  B,  6  days  \ 
C,  8  days;  and  D,  12  days.  They  received  $84  What 
was  each  man's  share? 


226  Chapter  Four. 

67.  A  man  has  768  hens,  which  is  \  more  than  he  had  last 
year.     How  many  had  he  then  ? 

68.  Two  trains  are  87£  miles  apart  and  running  toward 
each  other,  one  at  the  rate  of  50f  miles  an  hour,  and  the 
other  at  the  rate  of  20f  miles  an  hour.  How  far  apart  will 
they  be  in  half  an  hour  ? 

69.  If  35  men  earn  $  87.50  in  1  day,  how  much  will  50 
men  earn  in  10  days  ? 

70.  Multiply  9008  by  7080,  and  divide  the  product  by 
600. 

71.  What  is  the  difference  between  69  x  58.8  and  291  -*- 
0.97? 

72.  Find  6\%  of  19,712  miles. 

62i%  of  2768  yards. 
9^%  of  11,223,344  pounds. 

73.  What  is  the  interest  of  $  150  for  2  yr.  8  mo.  15  da., 
at  6%  per  annum. 

74.  Add :  25,037.45 ;  8,712.23 ;  9050.37 ;  815.25;  91,017.16  ; 
419.19;  2035.75;  15,025.55;  7079.13;  14026.27. 

75.  Add :  87.27 ;  43.75 ;  72.50 ;  39.75 ;  64.04;  58.94;  95.83 ; 
26.37;  75.96;  50.83;  39.49;  97.08;  62.62. 

76.  A  lot  of  land  containing  5250  square  feet  is  125  feet 
long.     What  is  the  perimeter  ? 

77.  A  man  spent  ^  of  his  money  for  a  house,  -^  for 
furniture,  -g^j-  for  horses,  and  ■§•  to  build  a  church.  What 
part  of  his  money  had  he  left  ? 

78.  Bought  10,752  cubic  feet  of  wood  at  $8J  a  cord. 
What  did  it  all  cost? 

70.   Change  *       ?  to  a  simple  fraction. 
80.   9|  times  £  of  56|  is  how  much  ? 


Review. 


227 


81.  What  is  the  cost  of  digging  a  cellar  27  feet  square 
and  9  feet  deep  at  25^  a  cubic  yard. 

82.  How  many  yards  of  fence  will  be  needed  to  enclose 
the  plot  of  ground  shown  in  the  following  diagram  ? 


5  rods 

4  rods 

to 

3 

1 

H 

19  rods 

TO 

0 

83.  The  above  field  was  originally  a  rectangle,  but  the 
owner  sold  one  piece  5  rods  by  3  rods,  and  a  second  piece  3 
rods  by  7  rods.  How  many  square  rods  did  it  contain  at 
first  ?     What  is  its  present  area  ? 

84.  Calculate  the  number 
of  square  yards  in  the  field 
shown  in  the  accompanying 
diagram. 


24  yds.. 


€ 


24  yds. 


85.  A  man  buys  a  piece  of 
ground  300  feet  long,  150  feet 
wide.  He  builds  a  house,  50  feet  by  30  feet,  and  a  shed  12 
feet  by  13  feet.  How  many  square  feet  will  he  have  left 
for  a  garden  ? 


228 


Chapter  Four. 


86.  The  owner  of  a  piece  of  ground  250  feet  long,  200  feet 
wide,  takes  10  feet  from  each  side  to  make  a  gravel  walk, 
and  uses  the  remainder  for  a  garden.  Give  the  dimensions 
of  the  garden  and  its  area  in  square  feet?  How  many- 
square  feet  in  the  whole  piece  of  ground?  How  many 
square  feet  are  taken  up  by  the  walk  ? 

87.  How  many  square  feet  of  flagging  would  be  required 
for  a  sidewalk  10  feet  wide  outside  a  lot  250  feet  long,  200 
feet  wide  ? 


88.  If  a  piece  of  carpet  is  27  inches  wide,  and  contains 
48  square  yards,  how  long  is  it  ? 

89.  I   have  bought  24  yards  of  dress  goods,  27  inches 
wide.     How  many  square  yards  does  the  piece  contain  ? 

How  many  yards  of  lining  32  inches  wide  will  contain 
the  same  number  of  square  yards  ? 


24  yards  long. 


?  yards  long. 


I  yd. 


18  sq.  yd. 


18  sq.  yd. 


I  yd. 


CHAPTER  V. 

PAGES 

Percentage 229  to  276 

Finding  Percentage,  Base,  Rate  ;  Commission,  Insur- 
ance, Duties,  Taxes,  Profit  and  Loss,  Commercial 
Discount,  Interest,  Partial  Payments,  Bank  Discount, 
Interest  by  Aliquot  Parts. 

Denominate  Numbers 277  to  291 

Reduction  Descending  and  Ascending,  Addition,  Sub- 
traction, Multiplication,  Division,  Review. 

Review  of  Simple  Numbers 291  to  309 

PERCENTAGE. 

302.  Preliminary  Exercises. 

Per  cent  means  hundredths.  Seven  per  cent  means  seven 
hundredths,  y^j-,  or  .07.     It  is  written  7%. 

How  many  hundredths  of  a  number  is  one  half  of  it? 
J  =  how  many  hundredths  ?     \?     $7     f?     -f? 

What  per  cent  of  a  number  is  the  half  of  it?   J?   J?   £  ? 

i?  i?  A?  A?  A?  A?  t**?  A*? 

What  per  cent  of  a  number  is  f  of  it?     J?     £?    f  ? 

A?  A?  A?  A?  A*? 

303.  1  per  cent  of  a  number  is  equal  to  what  fraction  of 
it?  3%?  5%?  9%?  10%?  15%?  20%?  25%? 
30%?    40%?     50%?     60%?     75%?    90%? 

304.  What  fractions  are  equal  to  the  following  ? 
12$%?  16|%?  33|%?  37$%?  6J%?  62$%?  66$%? 

87J%  ?   $%  ?  i%  ?  2$%  ?  $%  ? 

305.  3  times  a  number  is  what  per  cent  of  it  ?  2  J  times  ? 
li  times  ?   4$  times  ? 

229 


230  Chapter  Five. 

306.   Oral  Exercises. 

1.  Find  37J%  of  $24. 

37J  %  of  $  24  =  f  of  $  24,  or  $  9.     Ans.  $  9. 

2.  6%  of  150  bushels. 

1%  of  150  bushels  =  1.6  bushels  =  \\  bushels;  and  6%  is  6  times 
1£  bushels,  or  9  bushels.     Ans.  9  bushels. 

3.  81%  of  300  horses. 

81  %  of  100  horses  =  81  horses  ;  of  300  horses  it  is  3  times  81  horses, 
or  243  horses.     Ans.  243  horses. 

In  examples  2  and  3  the  pupil  should  be  led  to  see  that  he  can  point 
off  two  decimal  places  in  the  multiplicand  instead  of  in  the  multiplier  ; 
without  changing  the  result.  The  above  analyses  are  suggestive  merely. 
The  form  given  in  the  third  example  is  to  furnish  an  explanation  for 
the  use  of  3  as  a  multiplier. 

4.  Find  37|%  of  1  gallon. 

37 1  %  of  1  gal.  =  f  gal.  =  3  pt.  =  1  qt.  1  pt.,  Ans. 

5.  Find  121%  of  1  gallon  15.  66f%  of  66  horses 

6.  371%  of  $24  16.  l'6f%oflyard 

7.  33  J  %  of  81  cows  17.  81%  of  $300 

8.  6%  of  150  pounds  18.  2^%  of  80  sheep 

9.  4%  of  125  bushels  19.  40%  of  $2.50 

10.  62£%  of  1  peck  20.  20%  of  65  rods 

11.  4}%  of  $200  21.  10%  of  15  pounds 

12.  99%  of  200  gallons  22.  3J%  of  $60 

13.  \%  of  $640  23.  £%of$72 

14.  \%  of  800  yards  24.  lJ%of$96 

The  skilful  teacher  will  appreciate  the  importance  of  rapid  work, 
and  will  gradually  shorten  the  time  to  be  given  to  a  class  for  the  solu- 
tion of  an  oral  example.  She  will  also  vary  her  methods  of  conducting 
the  recitation,  so  as  to  keep  up  the  interest  of  the  pupils. 


Percentage. 


231 


TO  FIND  THE  PERCENTAGE. 

307.   "Written  Exercises. 

1.   Find  6%  of  $91.50. 
Multiply  the  base,  $91.50,  by  the  rate,  6,  ex 


pressed   as  hundredths.     The  result,  •$  5.49,   is 
called  the  percentage. 


$91.50 
x.06 
$5.4900    Arts. 


To  find  the  percentage,  multiply  the  base  by  the  rate  expressed 
as  hundredths. 


2.  331%  of  $28.80. 

While  the  rule  is  the  same,  to  multiply  $  28.80 
by  .38^,  the  pupil  should  not  fail  to  change  33£ 
hundredths  to  one-third. 

3.  \%  of  $1240. 

i°/o  =  jhj'  Divide  by  800  by  cancelling  the 
two  ciphers  in  the  divisor  and  making  two  deci- 
mal places  in  the  dividend. 


3)$  28.80 

$9.60     Ans. 


12.40/ 


$1.55  Ans. 


4.   41%  of  $92.40. 

$92.40  x  .04£. 

5.   450%  of  $92.40. 

$92.40  x  4.5. 

6.    12%  of  $37.50 

14. 

860%  of  $38 

7.   20%  of  $51.60 

15. 

\%  of  $2496 

8.    1400%  of  $89.70 

16. 

25%  of  $52.36 

9.    12i%  of  $73.28 

17. 

60%  of  $33.30 

10.   131%  of  $27.60 

18. 

8%  of  $19.50 

11.   6|%  of  $25.60 

19. 

6|%  of  $47.40 

12.   3£%  of  $47.40 

20. 

12%  of  $62.50 

13.    5£%  of  $29.50 

21. 

4i%  of  $  71.50 

232  Chapter  Five. 

22.  40%  of  $28.30  26.  75%  of  $59.20 

23.  160%  of  $39.40  27.  87|%of$392 

24.  84%  of  $23.75  28.  93f  %  of  $496 

25.  66f%  of  $825  29.  |f%0f$496 

Suggestion.  — The  teacher»should  have  a  preliminary  sight  lesson 
on  these  examples  before  giving  them  out  for  written  solution. 

TO  FIND  THE  BASE  OR  THE  RATE. 

308.   Preliminary  Exercises. 

1.  40  is  one-half  of  what  number  ? 

2.  40  is  .5  of  what  number? 

3.  40  is  50%  of  what  number  ? 

4.  40  is  what  part  of  80  ? 

5.  40  is  what  decimal  of  80  ? 

6.  40  is  how  many  hundredths  of  80  ? 

7.  40  is  what  per  cent  of  80  ? 

8.  26  is  what  per  cent  of  65  ? 

26  is  ff  of  65.  The  fraction  § £  equals  £,  or  40  hundredths.  Ans. 
40  per  cent. 

9.   26  is  40  per  cent  of  what  number  ? 

If  40  hundredths  of  a  number  is  26,  the  number  equals  26  divided 
by  40  hundredths,  or  26  -f-  .40.    Ans.  65. 

To  find  the  base,  divide  the  percentage  by  the  rate  expressed 
as  hundredths.  To  find  the  rate,  divide  the  percentage  by  the 
base,  expressing  the  result  in  hundredths. 

Another  method  of  finding  the  base  or  the  rate  is  suggested 
in  the  illustrative  examples  on  the  next  page,  which  give 
young  pupils  an  introduction  to  the  equation,  a  powerful 
instrument  in  mathematical  investigation. 


Percentage.  233 

.  Written  Exercises. 

1.  What  per  cent  of  65  is  26  ? 

This  means,  how  many  hundredths  of  65  will  equal  26  ?  which  may 
be  expressed  in  the  following  form  : 

65  x  —  =  26. 
100 

The  rate  being  required,  the  foregoing  may  be  written  as  follows  : 

65x  — =  26,  or  ^  =  26. 
100  100 

This  is  called  an  equation.  To  solve  the  equation,  that  is,  to  obtain 
the  value  of  r,  the  general  method  is  to  clear  the  equation  of  the  frac- 
tion by  multiplying  both  sides  by  the  denominator  of  the  fraction,  100. 
This  gives  65  r  =  2600,  or  65  times  r  equals  2600.  r,  therefore,  is  equal 
to  2600  divided  by  65.  Ans.  40  per  cent. 

Proof.  —  65  x  40  hundredths  =  26. 

2.  40  per  cent  of  what  number  equals  26  ? 

&Xi°-  =  26,  or  ^  =  26. 
100         '        100 

Clearing  of  fractions,  40  b  =  2600  ;   b  =  65,  Ans. 

Proof.  —  40  %  of  65  =  65  x  .40  =  26. 

3.  75  per  cent  of  a  number  is  42.    What  is  the  number  ? 

78%  =  I 

bx—  =  42,  or  ^=42. 
100  4 

Clearing  of  fractions,  3  6  =  168  ;  b  =  56,  Ans. 

4.  What  number  is  15  per  cent  of  84  ? 

p  =  15  hundredths  of  84. 

5.  24  is  18  per  cent  of  what  number  ? 

6.  27  per  cent  of  a  number  is  81.   What  is  the  number  ? 


234  Chapter  Five. 

7.  A  boy  spelled  correctly  20  words  of  25  given  out 
What  per  cent  of  the-  words  did  he  spell  correctly  ? 

Note.  — 25  is  the  base,  20  is  the  percentage  ;  required  the  rate. 

8.  132  is  120  per  cent  of  what  number  ? 

9.  -J-  per  cent  of  a  number  is  23.     What  is  the  number  ? 

10.  f  =  what  per  cent  of  -f  ? 

ix  _?!  =  §.      Cancelling,  -!-=-?. 
5      100      5  b    125     5 

Clear  of  fractions  by  multiplying  both  terms  of  the  equation  by  125. 

Prove  the  correctness  of  your  answer. 

310.  To  clear  an  equation  of  fractions,  multiply  both  terms 
of  the  equation  by  the  least  common  denominator  of  the 
fractions. 

11.  i  is  what  per  cent  of  f  ? 

12.  f  is  what  per  cent  of  £  ? 

5     100     4 '      '  '  125     4 

13.  3J  is  what  per  cent  of  §  ? 

14.  What  per  cent  of  $  389.50  is  $  124.64  ? 

15.  $  174.04  is  95%  of  what  sum  of  money  ? 

16.  f%  of  a  number  is  81.     What  is  the  number  ? 

ifa  of  6  =  81. 

17.  984  is  133£%  of  what  number  ? 

18.  What  number  increased  by  33£%  of  itself  equals 
984? 

Let  n  represent  the  number. 

Then  n  +  -  =  984  ;  i.  e.  —  =  084. 

Clearing  of  fractions,  4  n  =  984  x  3  =  2952.    n  =  738,  Am. 
Proof.  —  738  +  33$  %  of  738  =  738  +  246  =  984. 

19.  What  number  increased  by  25%  of  itself  equals  85? 


Percentage.  23  5 

311.   Oral  Exercises. 

1.  3  is  what  part  of  6  ? 

2.  3  is  what  decimal  of  6  ? 

3.  3  is  how  many  hundredths  of  6  ? 

4.  3  is  what  per  cent  of  6  ? 

5.  6  is  what  per  cent  of  3  ? 

6.  What  number  is  50%  of  6? 

7.  3  is  50%  of  what  number? 

8.  2  is  what  %  of  100  ? 

9.  2  is  what  %  of  200  ? 

10.  What  number  is  5%  of  100  ? 

11.  What  %  of  20  is  1? 

12.  4  is  what  %  of  200? 

13.  3  is  \°lo  of  what  number  ? 

14.  What  per  cent  of  9  is  20J  ? 

~  =  20^  ;  9  b  =  20£  hundred  ;  b  =  2\  hundred  =  225,  Ans. 
J.0U 

15.  What  number,  increased  by  ^  of  itself,  equals  10  ? 

16.  What  number,  increased  by  25  %  of  itself,  equals  20  ? 

17.  65  diminished  by  5%  of  itself  equals  what  ? 

18.  Buying  price  $  100,  selling  price  $  112.50.     Gain  %  ? 

19.  Cost  $  80,  profit  20%.     Selling  price  ? 

20.  What  principal  will  give  $  30  yearly  interest  at  6%  ? 

21.  A  man  had  $  600  in  the  bank.     He  drew  out  16f  per 
cent  of  it.     How  many  dollars  remained  in  the  bank  ? 

22.  A  lost  40  per  cent  of  his  money,  and  had  $  750  left. 
How  much  had  he  at  first  ? 


236  Chapter  Five. 

23.  If  I  am  compelled  to  lose  12^%  on  damaged  goods, 
how  must  I  sell  those  that  cost  me  $  5.60  ? 

24.  A  man  put  $15,  which  was  16|%  of  his  month's 
salary,  in  the  bank.     What  was  his  month's  salary  ? 

25.  If  each  boy  eats  60%  of  a  loaf  of  bread,  how  many 
boys  will  eat  6  loaves  ? 

Note.  —  In  the  solution  of  the  foregoing  oral  problems,  pupils 
should  not  be  compelled  to  use  the  method  suggested  for  the  written 
exercises. 

REVIEW. 
312.  Written  Problems. 

1.  A  man  receives  from  a  bank  4%  a  year  as  interest 
on  money  he  has  in  the  bank.  If  his  interest  for  a  year  is 
$  60,  how  much  money  has  he  in  the  bank  ? 

2.  A  city  had  a  population  of  4500  at  the  end  of  1903. 
The  population  at  the  end  of  1904  was  1080  greater.  What 
per  cent  did  the  population  increase  during  the  year  ? 

1080  =  what  per  cent  of  4500  ? 

3.  A  person  who  sold  an  article  for  25%  more  than  its 
cost,  received  $  85  for  it.     What  was  the  cost  ? 

Cost  +  £  cost  =  $  85. 

4.  A  person  receives  $  45  annual  interest  on  $  1000. 
What  rate  per  cent  does  he  receive  ? 

5.  A  farmer  sold  16J  per  cent  of  his  sheep,  and  had  75 
remaining.     How  many  had  he  at  first  ? 

6.  A  clerk  has  an  income  of  $1100  per  annum.  He 
pays  20  per  cent  of  it  for  board,  1£  per  cent  for  washing, 
2  per  cent  for  incidentals,  15  per  cent  for  clothing,  9  per 
cent  for  other  expenses,  and  loses  in  various  ways  50  per 
cent  of  the  amount  then  remaining.  What  sum  does  he 
have  left  ? 


Percentage.  237 

7.  What  per  cent  of  a  school  is  boys,  and  what  per  cent 
girls,  there  being  640  of  the  former  and  560  of  the  latter  ? 

8.  What  per  cent  of  9.075  is  24.2  ? 

9.  How  large  a  sale  must  a  merchant  make,  at  a  profit 
of  15  %,  that  his  gain  may  be  %  3750  ? 

10.  A  coal  dealer  bought  25,784  tons  of  coal  at  %  5  a  ton. 
He  sold  40%  of  it  at  %  7,  20%  of  it  at  $8.50,  and  the  remain- 
der at  %  4.50.     How  much  did  he  gain  ? 

11.  A  man  shipped  600  barrels  of  flour,  and  lost  16f%  of 
it  by  storm ;  he  sold  75%  of  the  remainder.  What  per  cent 
of  the  whole  remained  ? 

12.  66f  %  of  200  bushels  is  2£%  of  how  many  bushels  ? 

13.  If  corn  selling  for  21^  a  bushel  more  than  cost  gives 
a  profit  of  30%,  what  did  it  cost? 

14.  \  +  I  of  a  number  is  what  per  cent  of  it? 

15.  A  boy  deposited  $  15  in  bank.  This  was  30  per  cent 
of  what  he  had  in  bank  before  making  this  deposit.  What 
had  he  there  after  this  deposit  ? 

16.  A  man  can  do  a  certain  work  in  18f  days.  What  per 
cent  of  it  can  he  do  in  6|  days  ? 

17.  A  man  spent  30  per  cent  of  his  money  for  clothes,  20 
per  cent  for  rent,  and  had  $  75  left.     What  rent  did  he  pay  ? 

18.  What  is  the  difference  between  £  per  cent  of  $  15,000 
and  50  per  cent  of  $  15,000  ? 

19.  A  pole  extended  into  the  mud  5f  feet;  33^%  of  its 
length  was  in  the  river  and  25%  of  it  in  the  air.  What  was 
the  length  of  the  pole  ? 

20.  There  were  984  patients  in  a  certain  hospital,  classi- 
fied as  follows :  369,  pulmonary  diseases ;  246,  nervous  dis- 
eases ;  123,  diseases  of  heart ;  and  246,  various  other  diseases. 
Give  the  per  cent  of  each  class. 


238  Chapter  Five. 

APPLICATIONS  OF  PERCENTAGE. 

313.  Commission.  —  The  term  per  cent  occurs  in  many- 
business  transactions.  A  person  who  sells  goods  for  another 
receives  a  certain  per  cent  of  the  amount  he  obtains  for  the 
goods,  as  a  commission.  A  person  who  buys  goods  for 
another  is  paid  a  commission,  which  is  a  certain  per  cent  of 
the  cost  of  the  goods.  A  person  who  collects  money  for 
another  is  paid  a  commission  of  a  certain  per  cent  of  the 
amount  collected. 

Commission  is  a  percentage  paid  to  an  agent  for  his  services. 

314.  Insurance.  —  The  owner  of  property  who  desires  to 
be  insured  for  a  definite  sum  pays  some  per  cent  of  this 
sum  for  the  insurance.  The  amount  he  pays  is  called  the 
premium.  The  document  given  by  the  insurance  company 
as  a  receipt  is  called  a  policy.  It  states  the  agreement  of 
the  company  to  pay  the  owner  of  the  property  a  sum  equiva- 
lent to  the  loss  sustained,  provided  that  it  does  not  exceed 
the  sum  for  which  the  owner  is  insured.  Thus,  the  owner 
of  a  house  valued  at  $  5000  may  insure  it  against  fire  for 
$4000.  If  the  house  is  injured  to  the  extent  of  $4000  or 
less,  the  owner  receives  from  the  company  the  amount  of 
the  loss. 

Insurance  is  a  contract  by  which  one  party  agrees  to  pay  to 
another  a  specified  sum  in  case  of  loss  or  damage. 

Note.  —  The  teacher  should  show  pupils  an  insurance  policy,  and 
read  the  contract  made  by  the  company  as  expressed  therein. 

315.  Duties.  — The  United  States  government  collects  from 
the  importers  of  certain  classes  of  goods  a  stated  percentage 
of  the  value  of  the  goods.     This  charge  is  called  a  duty. 

Duties  are  taxes  on  imported  goods. 

Note.  —  Some  duties  are  based  upon  a  certain  rate  per  square  yard, 
per  pound,  etc. 


Applications  of  Percentage. 


*39 


316.  Taxes.  —  For  the  expenses  of  maintaining  a  city, 
property  owners  pay  a  certain  percentage  of  the  valuation 
of  their  property  as  determined  by  the  proper  officials.  The 
money  thus  collected  from  the  owner  is  called  a  tax.  The 
value  fixed  by  the  authorities  is  called  the  assessed  value, 
which  is  generally  somewhat  less  than  the  actual  value. 

A  tax  is  a  sum  of  money  levied  on  persons  or  property  for 
public  purposes. 

Note.  —  In  many  places  the  tax  rate  is  fixed  at  so  many  thou- 
sandths of  the  assessed  value. 


The  following  ten  oral  and  twenty  written  problems  in- 
volve no  new  principles.     The  general  formula  b  x  -^-  =p 

is  applicable  to  each  of  them.     The  accompanying  statement 

shows  the  base  on  which  the  percentage  is  calculated  in 

certain  classes  of  examples;   also  the  name  given  to  the 

percentage. 

Base 

JYalue  of  goods  bought 

[      or  sold,  etc 

f  Sum  for  which  property 
is  insured  .... 
Assessed  value  of  prop- 
erty       

Duties     ....  Value  of  goods  imported         Duty 


Commission  . 


Insurance 


Taxes 


Percentage 
Commission 
Brokerage 

Premium 
Taxes 


317.   Oral  Problems. 

1.  An  agent  collected  a  bill,  and  sent  to  his  employer 
the  amount,  less  2J%  commission.  If  his  commission  was 
$  1.60,  how  much  did  he  remit  to  his  employer  ? 

2.  My  house,  worth  $  12,000,  is  insured  for  J  of  its  value, 
at  \°lo'     What  premium  do  I  pay  ? 


24°  Chapter  Five. 

3.  A  man  collected  a  bill  of  $  300  for  me,  at  |%  com- 
mission.    How  much  was  his  commission  ? 

4.  Mr.  Eastman  collects  bills  for  me,  and  I  pay  him 
12  J%.  He  pays  over  to  me  §56.  How  much  did  he 
collect  ? 

5.  What  is  the  premium  for  insuring  $  3600  on  my  house 
at  |%  ? 

6.  What  will  it  cost  to  insure  a  house  worth  $  5000,  at 
\°/o  premium  ? 

7.  Eind  the  duty  at  35%  on  goods  valued  at  $  2000. 

8.  My  taxes  for  1904  are  $  175.     The  rate  is  If  per  cent. 

What  is  the  assessed  value  of  my  property  ? 

9.  My  agent  collects  the  yearly  rent  of  my  house,  and 
retains  $  15,  the  amount  of  his  commission  at-  2\  per  cent. 
Eor  how  much  does  the  house  rent  per  year  ? 

318.   Written  Problems. 

1.  How  much  insurance  does  a  man  receive  for  $  12.50 
when  the  rate  is  2|%  ? 

2.  An  importer  paid  duties  amounting  to  $386.75.  If 
the  duty  was  25%  of  the  cost  of  the  goods,  what  was  their 
cost? 

3.  A  collector  deducts  2J%  commission,  and  returns  to 
his  employer  $  745.68.     How  much  did  he  collect  ? 

Let  x  represent  the  sum  collected.     Then  2£%  of  se,  or  — ,  will  repre- 

of)  ~  40 

sent  the  commission  ;  and  x ,  or  -^—,  will  represent  the  amount 

returned  to  the  employer. 

^  =  745.68. 
40 

Clearing  of  fractions  :      39s  =  29,827.20. 

a;  =  764.80.    Ans.  $764.80. 

It  will  be  noted  that  only  abstract  numbers  are  used  in  an  equation, 
the  denomination  being  supplied  in  the  answer. 


Applications  of  Percentage.  241 

4.  The  tax  rate  of  a  certain  city  is  lf%  upon  the 
assessed  value  of  property.  If  this  value  is  75%  of  the 
actual  value,  how  much  taxes  does  Mr.  Smith  pay  upon  a 
house  and  lot,  the  actual  value  of  which  is  $  24,000  ? 

5.  The  tax  on  an  assessment  of  $8500  is  $48.45, 
Eequired  the  rate  on  $  1000  of  assessment. 

6.  Find  the  amount  of  an  agent's  sales,  when  his  com- 
mission at  5  per  cent  amounts  to  $  37.65. 

7.  An  agent  buying  wheat  is  offered  a  commission  of  4^ 
per  bushel,  or  one  of  4^-  per  cent,  and  he  chooses  the  former. 
The  average  price  paid  per  bushel  is  91^.  Does  he  gain 
or  lose  by  his  choice,  and  how  much  per  bushel  ? 

8.  A  commission  of  $121.29  was  charged  for  selling 
$  1866  worth  of  goods.     What  was  the  rate  of  commission  ? 

9.  A  man  insured  his  house  for  $  6500,  his  store  for 
$3500,  and  his  goods  for  $7000,  at  £%.  What  did  his 
insurance  come  to? 

10.  If  a  piece  of  property  is  taxed  $  28.60,  at  a  tax  rate 
of  -f  of  one  per  cent,  what  is  the  assessed  value  of  the 
property  ? 

11.  A  house  valued  at  $  24,000  was  insured  for  two-thirds 
of  its  value,  at  f  %.     What  is  the  premium  ? 

12.  An  agent  collected  20%  of  an  account  of  $  750,  charg- 
ing 4%  commission.  What  was  his  commission,  and  what 
sum  should  he  have  paid  over  ? 

13.  Paid  $27  for  an  insurance  policy  on  my  house.  If 
the  rate  is  f  %,  for  how  much  is  my  house  insured  ? 

14.  My  agent  collected  80  per  cent  of  a  debt  of  $  4500, 
and  charged  1\  per  cent  commission.  What  amount  should 
he  pay  me  ? 


242  Chapter  Five. 

15.  A  farmer  bought  6  cows  through  an  agent.  He  sent 
$  525.30  to  pay  for  the  cows  and  a  commission  of  3%.  How 
much  did  each  cow  cost  ? 

16.  What  will  be  a  broker's  commission,  at  2|%,  for  sell- 
ing a  farm  of  673  acres  @  $52  per  acre  ? 

17.  If  the  tax  rate  is  $13.80  on  $1000,  what  is  the 
assessed  value  of  property  that  pays  a  tax  of  $  144.90  ? 

18.  A  house  is  insured  for  -|  of  its  value  at  -J%.  The 
annual  cost  (premium)  is  $  8.40.  What  is  the  value  of  the 
house  ? 

Let  x  represent  the  value.  Then  — ^  x  — ,  or  — — ,  will  represent 
4.  .  3       800        1200 

the  premium. 

7  x 
The  equation  becomes  — —  =  8.40. 

19.  What  will  be  the  taxes  on  a  house  worth  $48,000 
and  assessed  at  -|  of  its  value,  the  tax  rate  being  $  18.50  per 
$  1000  of  assessed  value  ? 

20.  A  commission  merchant  receives  2\°fo  commission  for 
buying  grain  for  a  customer.  The  cost  of  the  grain  and  his 
commission  amount  to  $  4223.  How  much  does  the  grain 
cost? 

Let  x  represent  the  cost  of  the  grain  ;  —  will  be  the  commission. 

21.  An  importer  paid  $134.40  duties  on  imported  goods 
valued  at  $  384.     Find  the  rate. 

22.  What  is  the  duty  in  United  States  money  on  glass 
ware  valued  at  1500  francs,  the  rate  being  60%,  and  the 
franc  being  worth  19.3  cents  ? 

23.  Find  the  duty  on  a  gross  of  scissors,  valued  at  $2.50 
per  dozen,  the  rate  being  75  cents  per  dozen  and  25%  on 
the  value. 


Profit  and  Loss.  243 

PROFIT  AND  LOSS. 

In  determining  the  rate  per  cent  of  gain  or  loss  on  goods 
sold,  the  buying  price  of  the  goods  is  taken  as  the  base. 

319.   Oral  Problems. 

1.  What  is  the  gain  per  cent  on  sugar  bought  at  5  cents 
per  pound  and  sold  at  6  cents  per  pound  ? 

Profit  \f,  which  is  one-fifth  of  buying  price,  or  20%. 

2.  By  selling  a  house  for  $  3500, 1  lose  $  500.     What  is 
my  loss  per  cent  ? 

The  loss,  $  500,  is  what  per  cent  of  the  cost,  $  4000  ? 

3.  By  selling  a  lot  for  $  1000,  Mr.  Jones  loses  20  per 
cent.     What  did  the  lot  cost  ? 

The  selling  price,  $  1000,  is  four-fifths  of  the  cost. 

4.  Find  the  cost  of  an  article  which  was  sold  for  $  60, 
at  a  loss  of  70%. 

5.  If  I  buy  a  dozen  pencils  at  2^  each,  and  sell  at  3fi 
each,  what  is  the  gain  per  cent  ? 

6.  A  saddle  was  sold  for  $  18,  which  was  12 J- %  more 
than  the  cost.     How  much  did  it  cost  ? 

7.  What    %    is   gained   on  goods   sold  at   double   the 
cost? 

8.  Sold  flour  at  a  profit  of  $  2,  and  gained  25</0.     What 
was  the  cost  per  barrel  ? 

9.  What  is  the  %  of  gain,  when  boots  which  cost  $  2  a 
pair  are  sold  for  $2.50? 

10.  What  per  cent  is  lost  in  buying  potatoes  at  80^  a 
bushel,  and  selling  them  at  60  ^  a  bushel  ? 

11.  If  I  buy  butter  at  30^  a  pound,  how  much  per  cent 
do  I  gain  by  selling  it  at  36  j  a  pound  ? 


244  Chapter  Five. 

320.  Written  Exercises. 

Find  the  profit  or  the  loss,  and  the  selling  price  : 

1.  Cost  $1876;  gain  15%. 

Gain  =  15  per  cent  of  $  1876.    Selling  price  =  cost  +  gain. 

2.  Cost  $36.75;  loss  20%. 

3.  Cost  $1012.50;  gain  16|%. 

4.  Cost  $875;  loss  5%. 

5.  Cost  $934.56;  gain  12£%. 

^  Find  the  profit  or  the  loss  per  cent. 

6.  Cost  $600;  selling  price  $618. 

$18=  ?%of  $600. 

7.  Cost  $  1203;  selling  price  $802. 

8.  Cost  $86.20;  selling  price  $  73.27. 

9.  Cost  $908.40;  selling  price  $1090.08. 

10.  Cost  $84;  selling  price  $78.75. 

11.  Selling  price  $78.75;  loss  $5.25. 
Note. —Cost  =  $78.75 +  $5.25  =  $84 

12.  Selling  price  $  150;  gain  $  25. 
Use  the  cost  ($  150  -  $25)  as  the  hase. 

13.  Selling  price  $831.25;  loss  $43.75. 

14.  Selling  price  $  1051.38 ;  gain  $  116.82. 

15.  Selling  price  $843.75;  loss  $168.75. 

Find  the  cost,  and  the  profit  or  loss : 

16.  Selling  price  $  468.75 ;  gain  25%. 

5x 

Representing  the  cost  by  x,  the  selling  price  is  —-• 

^  =  468.76. 


Profit  and  Loss.  245 

17.  Selling  price  $73.84 5  loss  20%. 

18.  Selling  price  $1646.08;  gain  33 \% 

19.  Selling  price  $204;  loss  15%. 

20.  Selling  price  $66.30;  gain  4%. 

21.  A  man  buys  a  horse  for  $  275,  and  sells  it  at  a  profit 
of  20  per  cent.     How  much  does  he  gain  ? 

22.  A  cow  is  sold  for  $75,  on  which  the  profit  is  $15. 
What  is  the  gain  per  cent  ? 

23.  A  lot  is  sold  for  $  960,  which  is  20  per  cent  more  than 
it  cost.     Find  the  cost  of  the  lot. 

24.  Tea  that  costs  32  f  per  pound  is  sold  for  48  £  What 
is  the  gain  per  cent  ? 

25.  A  man  buys  a  horse  for  $  175  and  sells  it  for  $200. 
What  per  cent  does  he  gain  ? 

26.  What  per  cent  was  lost  on  a  horse  that  cost  $200, 
and  that  was  sold  at  a  loss  of  $  25  ? 

27.  What  is  the  selling  price  of  dress  goods  costing  ?&\i 
per  yard,  on  which  a  profit  of  VZ\  per  cent  is  made  ? 

28.  Sold  a  coat  for  $  33.60,  thereby  losing  16  per  cent. 
What  was  its  cost  ? 

29.  How  much  did  I  gain  on  a  house  for  which  I  paid 
$8760,  my  profit  being  2\  per  cent? 

30.  A  man  paid  for  a  house  $4500,  and  for  repairs  $  150, 
and  then  sold  it  for  18%  above  the  entire  cost.  What  did 
he  receive  for  it  ? 

31.  To  make  15  per  cent  profit,  what  must  goods  be 
marked  that  cost  96  cents  per  yard  ? 

32.  Goods  costing  96  cents  per  yard  are  marked  at  25% 
advance,  what  per  cent  is  gained  if  they  are  sold  10  %  below 
the  marked  price  ? 

33.  Find  the  per  cent  of  profit  on  apples  bought  at  $  1.25 
per  bushel,  and  sold  at  25  cents  per  half  peck. 


246 


Chapter  Five. 


COMMERCIAL  DISCOUNT. 

321.  Wholesale  dealers  in  certain  classes  of  goods  allow 
to  purchasers  of  large  quantities  a  deduction  from  the  prices 
printed  in  their  catalogues.  This  is  called  a  trade  discount 
or  commercial  discount.  A  discount  for  prompt  payment  is 
also  frequently  allowed.  The  following  bill  contains  a  trade 
discount  and  a  discount  for  cash : 

Philadelphia,  Jan.  17,  1904. 
The  Ocean  Bathing  Suit  Co. 

Terms:  Cash  less  5  per  cent.  Sold  to  Mr.  J.  H.  HAAREN. 


12%  doz.  Suits 


918 
less  15% 


Cash  less  5°J0 

Rec'd  Paym% 
Jan.  17,  1904, 

0.  B.  S.  CO. 

per  M.  M. 


§225 

— 

S3 

75 

$191 

25 

9 

56 

$181 

69 


1.  Make  out  a  bill  for  16  gross  of  roman  candles  at 
$26.75  per  gross,  less  60%. 

2.  On  a  bill  of  goods  amounting  to  $583.40,  a  discount 
of  5%  is  given  for  cash.     What  is  the  amount  paid  ? 

3.  Sept.  1,  1905,  Mr.  Maxwell  bought  tea  amounting  to 
$  1876.50.  If  5%  is  deducted  for  payment  within  ten  days, 
how  much  should  he  pay  if  he  paid  the  bill  Sept.  9  ? 

4.  What  will  be  the  cost  of  15  cases  cocoa  @  $13.20 
each,  less  20%  ? 


Commercial  Discount.  247 

5.  Bought  5  gross  of  essence  of  lemon  at  50^  per  doz., 
less  5%.     What  is  the  amount  of  my  bill  ? 

6.  Find  the  cost  of  15  cases  of  chloride  of  lime,  50  lb. 
per  case,  at  9|-^  per  pound,  less  15%. 

7.  Find  the   cost  of  a  wagon,  the  catalogue  price  of 
which  is  $750,  the  discount  being  30%. 

8.  What  will  be  the  cost  of  goods  amounting  to  $  1837.60, 
on  which  there  is  allowed  a  discount  of  17^  %  ? 

9.  Find  the  net  cost  of  1630  yd.  silk,  invoiced  at  $  1.10 
per  yard,  less  16%  discount. 

Note.  —  The  amount  previous  to  the  deduction  of  the  discount  is 
known  as  the  gross  amount.  The  net  amount  or  the  net  cost  is  the 
sum  actually  due  after  the  deduction  of  the  discount. 

10.  What  is  the  cost,  in  francs,  of  843.72  meters  silk,  at 
5.75  francs  per  meter,  less  12%  ? 

11.  What  will  be  the  net  cost  of  a  bill  of  plated  ware 
amounting  to  $  84.75,  on  which  a  discount  of  33^  and 
10%  is  allowed? 

$  84.75 
1         !       no  ok  When  more  than   one  discount  is  given, 


56.50 


each  successive  discount  is  based  on  the  re- 
mainder left  after  the  deduction  of  the  previ- 


less  A-      5.65 

.    1  °_ ous  discount. 

Ans.  $  net. 

Note.  —  The  mark  %  is  generally  written  only  after  the  last  rate. 

12.  Find  the  difference  between  $  390  less  43^%  discount, 
and  $  390  less  33J  and  10%  discount. 

13.  An  army  fought  two  battles.  In  the  first  it  lost  15 
per  cent,  and  in  the  second  20  per  cent  of  the  original  num- 
ber, after  which  it  mustered  19,500  men.  What  was  the 
original  strength  of  the  army  ? 


248  Chapter  Five. 

14.  Find  the  net  cost  of  18,500  bags  at  $  4.40  per  M,  less 
60  and  10  and  5%. 

15.  What  is  the  net  cost  of  a  lot  of  musical  instruments 
amounting  to  $  1875.60,  on  which  a  discount  of  10,  5,  and 
2\%  is  allowed? 

16.  What  would  be  the  net  cost  of  the  same  articles,  if 
the  discount  were  2^-,  5,  and  10%  ? 

17.  Find  the  net  cost  of  the  same,  at  17|-%  discount. 

18.  Which  is  the  better  discount  for  the  buyer,  40  and 
10%  or  30  and  20  ?  What  will  be  the  difference  on  a  bill 
of  $  100  ? 

19.  $100  less  33  J  and  10%  discount  is  equal  to  what? 
What  per  cent  discount  is  33 J  and  10  %  equal  to  ? 

20.  $  100  less  10  and  33J%  is  equal  to  what  ? 

Note.  —  The  pupil  will  note  that  the  result  is  the  same  as  in 
Problem  19. 

21.  A  man  marks  an  article  $1.50,  and  sells  it  at  a  dis- 
count of  25%  from  the  marked  price.  If  the  article  cost 
him  90  ^,  what  is  his  gain  per  cent  ? 

22.  John  Jasper  &  Co.  sold  the  following  goods.  Make 
out  the  bill,  less  50  and  10  and  10  and  10%  discount. 

500  ^-pound  bags  at  $  1.00  per  M. 
1500  -i-pound  bags  at  1.20  per  M. 
3000  1-pound  bags  at  1.60  per  M. 
5500  Impound  bags  at  1.70  per  M. 
2000    2-pound  bags  at     2.00  per  M. 

Note.  —  In  making  out  large  numbers  of  bills  clerks  have  no  time 
for  unnecessary  words.    The  first  item  would  be  written  as  follows : 

600  \  lb.  Bags  $1.  .50 

the  words  "  at "  and  M  per  M  "  being  considered  unnecessary. 


Commercial   Discount.  249 

322.   Oral  Problems. 

1 .  A  piano,  marked  $  800,  is  sold  at  a  discount  of  25  and 
10%.     What  is  the  selling  price  ? 

2.  Bought  goods  amounting  to  $600,  less  5%  for  cash. 
What  is  the  net  cost  of  the  goods  ? 

3.  What  single  discount  is  50  and  10%  equal  to? 
Taking  f  100  as  a  base,  50  %  discount  deducts  f  50  and  leaves  $  50. 

10  %  deducts  $  5,  leaving  $  45.      The  total  deduction  is  $  55  ;  the 
single  equivalent  discount  is  55  %. 

4.  What  single  discount  is  30  and  30%  equal  to? 
Representing  the  base  by  x,  the  first  discount  is  30  %  of  x,  or  — — , 

leaving  — -.     The  second  discount  is  T37  of  — — ,  which  is  —^r.    The 

Qfk  91  x,^ 

two  discounts  are  —p-r  and  ■——,  which  make  a  total  of  — — ,  or  51  %  of 
.,      ,  1UO  1U0  100 

the  base. 

5.  Paid  $729  for  goods,  on  which  10%  was  allowed. 
What  was  the  "  gross  "  price  ? 

-  6.  How  much  will  be  paid  for  12  doz.  bottles  flavoring 
extract,  at  60^  per  dozen,  less  10%  ? 

7.  What  is  the  "list "  price  of  an  article  for  which  I  paid 
$  48,  after  a  discount  of  25%  was  deducted  ? 

Note.  — The  "  list  "  price,  "  gross  "  price,  or  "  catalogue  "  price  ia 
the  price  before  the  deduction  of  discounts. 

8.  What  is  the  net  price  of  an  article  catalogued  at  $  880, 
on  which  there  is  a  discount  of  75%  ? 

Note.  —  A  discount  of  75  %  from  the  list  price  means  that  the  net 
price  is  25  %  of  the  list  price.  Instead,  therefore,  of  finding  75  %  of 
$  880  and  deducting  it  from  $  880,  the  pupil  should  shorten  the  work 
by  taking  25%  of  $880. 

9.  75  is  25%  more  than  what  number? 

10.  Find  the  cost  of  a  wagon  "  catalogued  "  at  $  700,  the 
discount  being  30%. 

Note.  —  The  cost  is  70  %  of  $  700. 


250  Chapter  Five. 

INTEREST. 

323.  Preliminary  Exercises. 

1.  A  farmer,  needing  $1000  to  purchase  additional  land, 
borrows  the  money,  agreeing  to  repay  it  at  a  given  time 
with  6%  of  the  sum  for  each  year  he  has  the  use  of  it. 
This  annual  payment  of  $  60  for  the  use  of  $  1000  is  called 
interest.  The  $  1000  is  called  the  principal.  If  the  borrower 
repays  the  $  1000  at  the  end  of  two  years  and  also  $  120 
interest,  the  total  payment  of  $  1120  is  called  the  amount. 

2.  What  is  the  interest  on  $1000  at  6%  for  6  months  ? 

3.  What  is  the  amount  of  $1000  for  3  years  at  6%  ? 

4.  What  is  the  interest  on  $  1000  for  1  month  at  6%  ? 

5.  Taking  30  days  to  a  month,  find  the  interest  on 
$1000  for  15  days  at  6%. 

324.  "Written  Exercises. 

1.   Find  the  interest  on  $  750  at  6%  for  2  years  6  months. 

$750   Principal 
The  interest  for  1  year  A«  ^ 

.„,,,.,.,  X  .06  Rate. 

is  found  by  multiplying  the  gKnA 

•     •     1  atca  v   *i.  Interest  for  1  yr.  $45.00 

principal,  $750,  by  the  rate,  01      • 

6  hundredths,  and  this  prod-  mn*    ?  Time* 

uct  by  the  number  of  years, 

2*. 


22.50 


Interest  for  2  yr.  6  mo.  $  112.50 
The  foregoing  may  be  expressed  by  the  formula : 

Principal  x  ^^  x  Time  (in  years)  =  Interest 
100 

It    is    suggested 

that   the   work    be  ^^        3 

arranged     in     this  $             ±_      5      $2^5  =  fmM    ^ 

manner,  so  that  it  rr      J[fifl      JJ          2 

may   be   shortened  2 

by  cancellation. 


Interest.  251 


2.   What  is  the  interest  on  $84.75  at  4%  for  3  months 
6  days  ?  3  months  6  days  =  qq  days  =  96  year 

ooO 

The  100  in  the  divisor  is  can-  .0565        i         aa 

celled  by  removing  the  decimal  $.84/75  X  =7777  X  07577  =  $  .9040 

point  in  the  principal  two  places  qa 

to  the  left.  Ans^  go  centg        0 


3.   Find  the  interest  on  $394.50  for  2  years  7  months 

*  **%.  477 

$.394/50x   ?   xW     $188.1765 


24  days  at  4|%.  4?? 


4 


=  $47,041+.  ^ns.  $47.04. 


The  time  is  readily  changed  to  days  :  720  +  210  +  24  =  954. 

This  is  expressed  in  years  by  placing  360  in  the  divisor,  i.e.  below 
the  line.  The  three  ciphers  in  the  divisor  are  cancelled  by  moving 
the  decimal  point  in  the  principal  three  places  to  the  left. 

In  calculating  interest,  take  30  days  to  a  month,  12  months  to  a 
year. 

Find  the  interest  on 

4.  $308  at  5%  for  20  days.       6.    $720  at  7%  for  21  days. 

5.  $360  at  5%  for  33  days.       7.    $1000  at  5%  for  8  days. 

8.  $94.43  at  7%  for  2  mo.  3  da. 

9.  $464.75  at  6%  for  8  mo.  12  da. 
10.    $400  at  41%  for  1  yr.  1  mo.  1  da. 

325.   Amount  =  Principal  +  Interest 
Find  the  amount : 

1.    $  813,  from  April  19,  1902,  to  March  4,  1907,  at  6%. 
The  time  is  found  by  compound  subtraction.      1907       3       4 
4  yr.  10  mo.  15  da.  1902       4     19 

.271  4     10     15 

Interest  =  S-TO/xg    x  1755  =  %  m M 

m  x  m 
2 

Amount  =  $  813  +  $  237.80  =  $  1050.80,  Arts. 


252  Chapter  Five. 

2.  $  960,  from  Jan.  1,  1903,  to  Dec.  21, 1904,  at  4%. 

3.  $  27.84,  for  3  yr.  6  mo.  9  da.,  at  6%. 

4.  $  48.90,  for  17  da.,  at  6%. 

5.  $  144,  for  2  yr.  5  da.,  at  3f  %. 

6.  $  834.76,  for  15  mo.  27  da.,  at  4J%. 

7.  $  5760,  for  1  yr.  5  mo.  29  da.,  at  5%. 

8.  9  2346.50,  for  7  yr.  13  da.,  at  3%. 

9.  $  1892,  for  3  yr.  5  mo.,  at  7%. 

10.   $  150.40,  for  1  yr.  2  mo.  3  da.,  at  6%. 

326.   Interest-bearing  Demand  Notes. 

A  promissory  note  is  a  written  agreement  to  pay  a  stated 
sum  of  money  after  a  given  time  or  on  demand  to  a  certain 
person  with  or  without  interest.  The  person  signing  the  note 
below,  James  Dunne,  is  called  the  maker;  the  person  in  whose 
favor  it  is  drawn,  Charles  C.  Wise,  the  payee.  If  the  latter 
wishes  to  transfer  it  to  James  H.  Tully,  he  writes  on  the 
back  of  the  note :  Pay  to  the  order  of  James  H.  Tully ;  and 
underneath  he  signs  his  name.  This  is  called  an  indorse- 
ment in  full.  By  merely  signing  his  name  on  the  back, 
which  is  called  an  indorsement  in  blank,  Mr.  Wise  makes  it 
payable  to  any  person  holding  it.  The  effect  of  an  indorse- 
ment is  also  to  make  the  indorser  liable  in  the  event  of  the 
maker  refusing  to  pay. 

1.  San  Francisco,  Jan.  7,  1902. 

On  demand,  I  promise  to  pay  Charles  C.  Wise,  or  order, 
Seven  Hundred  Sixty-five  T4^j-  Dollars,  value  received,  with 
interest  at  6  per  cent. 

$765TVtf.  James  Dunne. 

•     How  much  money  will  be  required  to  pay  the  above  note, 
with  interest,  July  15,  1903  ? 


Interest.  253 

2.  A  demand  note,  dated  Sept.  25,  1902,  with  interest  at 
8%  from  date,  is  paid  Jan.  2,  1905.  How  much  was  due, 
the  face  of  the  note  being  $  750  ? 

3o  Find  the  amount  due  March  4,  1904,  on  a  note  for 
$  365.84,  dated  May  20, 1902,  with  interest  from  date  at  7%. 

4.  Find  the  amount  necessary,  Oct.  16,  1906,  to  pay  a 
note  of  $  1240,  with  interest  at  6%  from  Aug.  15,  1902. 

5.  An  interest-bearing  note  for  $  87.60  is  dated  April  3, 
1900.  How  much  is  due  on  it  for  principal  and  interest 
Jan.  2,1908?    Rate  4^%. 

327.    Oral  Problems. 

If  these  are  first  used  as  sight  problems,  an  opportunity  will  be 
afforded  to  develop  different  methods  for  solving  many  of  them. 

1.  Find  the  interest  on  $  300,  for  1  yr.  7  mo.,  at  4%. 

$  12  per  year  is  how  much  for  7  months  ? 

2.  On  $  60,  for  33  days,  at  6%. 

$  3.60  for  360  days  is  how  much  for  33  days  ? 

3.  On  $  120,  from  Jan.  1,  1903,  to  July  1,  1904,  at  5%. 

4-  How  long  will  it  take  $  100  to  produce  $  15,  interest 
at6%? 

5.  At  what  rate  per  cent  will  $  50  produce  $  6  in  2  years  ? 

6.  What  is  the  interest  on  $  300,  at  6%,  from  Feb.  1  to 
Feb.  21  ? 

7.  What  part  of  a  year  is  72  days  ? 

8.  Find  the  interest  at  4%,  for  90  days,  on  f  150. 

9.  On  1 240,  for  36  days,  at  5%. 

10.  What  is  the  amount  of  $  200,  for  3  yr.  1  mo.,  at  6%  ? 

11.  How  long  will  it  take  $  1  to  make  $  1  interest  at  5%  ? 

12.  How  long  will  it  take  any  sum  to  double  itself  at  6%  ? 

13.  How  long  will  it  take  $  14.90  to  double  itself  at  4%  ? 


254  Chapter  Five. 

PARTIAL  PAYMENTS. 

328.   United  States  Eule. 

.  When  the  maker  of  an  interest-bearing  note  pays  a  portion 
of  the  debt  represented  by  the  note,  the  money  is  applied  in 
the  first  place  to  the  payment  of  the  interest,  then  to  the 
reduction  of  the  principal. 

1.  Duluth,  Minn.,  Jan.  5,  1902. 

On  demand,  I  promise  to  pay  to  the  order  of  James  F. 
McGee  Three  Hundred  Dollars,  value  received,  with  interest 
at  7  per  cent. 

$300^.  J.  Eandolph  Page. 

Payments:  May  20,  1902,  $100;  Oct.  30,  1902,  $100; 
March  6,  1903,  $50. 

How  much  was  due  Jan.  5,  1904  ? 

Find  amount  of  $300  Jan.  5,  1902,  to  first  payment  May  20,  1902, 

4  mo.  15  da.  (by  compound  subtraction),  $307.88 

Deduct  first  payment,  100.00 

Balance  May  20,  1902,  $207.88 

Interest  on  $  207.88  to  Oct.  30,  5  mo.  10  da.,  6.47 

Amount,  $214.35 

Less  second  payment,  100.00 

Balance  Oct.  30,  1902,  $114.35 

Interest  on  $  114.35  Oct.  30  to  March  6,  4  mo.  6  da.,  2.80 

Amount,  $117.15 

Less  third  payment,  50.00 

Balance  March  6,  1903,  $67.15 

Interest  on  $67.15  March  6  to  Jan.  5,  9  mo.  29  da.,  3.90 

Due  Jan.  6,  1904,  $71.05 

Find  the  amount  of  the  principal  to  the  time  when  the  pay- 
ment or  the  sum  of  two  or  more  payments  equals  or  exceeds  the 
interest. 

From  this  amount  deduct  the  payment  or  sum  of  payments. 

Use  the  balance  then  due  as  a  new  principal,  and  proceed  as 
before. 


Partial  Payments.  2$$ 

2.  How  much  is  due  June  3,  1905,  on  a  demand  note  for 
$1200,  with  interest  at  6%,  dated  June  3,  1902,  bearing 
indorsements  of  payment  of  $  500,  Sept.  18,  1903 ;  $  600, 
Jan.  3,  1904? 

Note.  — Anything  written  on  the  back  of  a  document  is  called  an 
indorsement.  Payments  made  are  usually  written  on  the  back  of  the 
notes. 

3.  A  demand  note  for  $  &00,  bearing  interest  at  5  %,  was 
given  Feb.  18,  1902.  A  payment  of  $  250  was  made  May 
28,  1903 ;  one  of  $  150  was  made  Oct.  8,  1903.  How  much 
is  due  Jan.  23, 1905  ? 

4.  A  note  for  $2000,  with  interest  at  7%,  was  dated 
April  15,  1901.  Indorsements  were  made  as  follows :  $  50, 
Sept.  20, 1901 ;  $  100,  May  26, 1902 ;  $  1000,  June  20,  1903. 
How  much  is  due  Dec.  27,  1904  ? 

Face  of  note,  $2000.00 

Interest  from  April  15  to  Sept.  20,  1901,  5  mo.  5  da.,  60.28 

Amount  due  Sept.  20,  1901,  $2060.28 

If  the  $  50  payment  were  deducted,  and  interest  computed 
on  the  balance,  $2010.27,  the  maker  would  be  charged  in- 
terest on  $  10.27  more  than  the  face  of  the  note,  and  this  the 
law  does  not  allow.  Interest  is  taken  on  $  2000  until  next 
payment,  May  26,  1902,  8  mo.  6  da.,  95.67 

Amount  due  May  26,  1902,  $2155.95 

As  the  two  payments  are  not  large  enough  to  meet  the 
interest  now  due,  the  interest  is  again  computed  on  the 
original  $2000  from  May  26,  1902,  to  June  20,  1903,  1  yr. 
24  da.,  149.33 

Amount  of  $  2000  from  April  15,  1901  to  June  20,  1903,     $  2305.28 
Less  $  50  +  $  100  +  $  1000  (three  payments),  1150.00 

Balance  due  June  20,  1903,  $  1155.28 

Interest  on  $  1155.26  to  Dec.  27,  1904,  1  yr.  6  mo.  7  da.,         122.87 
Due  Dec.  27,  1904,  $  1278.16 


256  Chapter  Five. 

1 

5.  Albany,  N.Y.,  March  5,  1903. 
One  year  after  date,  I  promise  to  jpay  John  Harrigan,  or 

order,  Nine  Hundred  Dollars,  value  received,  with  interest 
at  six  per  cent. 

$  900^.  Andrew  T.  Sullivan. 

Indorsed  as  follows :  June  5,  1903,  $  10 ;  Sept.  5,  1903, 
$  50 ;  Jan.  5,  1904,  $  120.     What  was  due  March  8,  1904  ? 

6.  Alexandria,  La.,  June  19,  1903. 
On  demand  I  promise  to  pay  to  the  order  of  George  H. 

Dotzert,Two  Thousand  Four  Hundred  Fifty-four  -ftjfo  Dollars, 
value  received,  with  interest  at  6  per  cent. 

$2454£fc  Charles  W.  Lyon. 

The  following  payments  were  made:  July  5,  1903,  $450; 
Sept.  18, 1903,  $  700 ;  Oct.  25, 1903,  $  300.  Find  the  amount 
due  Jan.  2,  1904. 

329.  In  the  United  States  courts,  and  in  those  of  some  of  the  states, 
interest  for  a  portion  of  a  year  is  taken  by  days,  upon  the  basis  of  365 
days  to  the  year.  To  make  the  work  easier  for  the  pupils,  however, 
the  year  of  360  days  should  be  used  in  the  examples  given,  and  the 
time  between  dates  should  be  found  by  compound  subtraction. 

330.  Merchants'  Eule. 

The  merchants'  rule  is  frequently  used  where  all  the 
payments  are  made  within  a  year. 

The  interest  is  computed  on  the  face  of  an  interest- 
bearing  note  from  its  date  until  settlement,  and  interest  is 
allowed  on  all  credits  from  their  payment  until  settlement. 

The  exact  number  of  days  is  taken,  and  the  interest  is 
computed  on  the  basis  of  360  days  to  the  year. 

Boston,  Mass.,  June  19,  1903. 

On  demand,  I  promise  to  pay  Charles  R.  Buttrick,  or 
order,  Two  Thousand  Four  Hundred  Fifty-four  -^  Dollars, 
value  received,  with  interest  at  6  per  cent. 

$  2454^.  John  J.  P.  Fagan. 


ne, 

82454.75 

80.60 

$2535.35 

$  200.00 

6.03 

450.00 

11.78 

700.00 

12.37 

300.00 

3.45 

1683.63 

Partial  Payments.  257 

The  following  payments  are  endorsed  on  the  note :  July  5, 
1903,  $200;  July  29,  1903,  $450;  Sept.  18,  1903,  $700; 
Oct.  25,  1903,  $  300. 

Find  the  amount  due  Jan.  2,  1904. 

If  no  payments  had  been  made,  there  would  be  due, 

And  interest  from  June  19  to  Jan.  2,  197  days, 

Total  due, 

The  credits  are:  Payment  July  5,  1903, 

Interest  on  $  200,  July  5  to  Jan.  2,  181  days, 

Payment  July  29,  1903, 

Interest  on  $450,  July  29  to  Jan.  2,  157  days, 

Payment  Sept.  18,  1903, 

Interest  on  $  700,  Sept.  18  to  Jan.  2,  106  days, 

Payment  Oct.  25,  1903, 

Interest  on  $  300,  Oct.  25  to  Jan.  2,  69  days. 

Balance  due,  $  851.72 

Find  the  amount  of  an  interest-bearing  note  at  the  time  of 
settlement. 

Find  the  amount  of  each  credit  from  its  time  of  payment  to 
the  time  of  settlement ;  subtract  their  sum  from  the  amount  of 
the  note. 

331.   Written  Exercises. 

1.  A  note  for  $  500,  with  interest  at  6%,  is  dated  July 
25, 1904.  Payments  are  made:  $  100,  Sept.  18  ;  $  200,  Feb. 
5,  1905.     How  much  is  due  April  1,  1905? 

2.  Find  amount  due  Sept.  15,  1903,  on  a  demand  note  for 
$  1875,  with  interest  at  6  %,  dated  Jan.  18, 1903.  Payments 
of  $  1000  and  $  500  were  made  March  30  and  June  17, 
respectively. 

3.  June  12,  1904,  Robert  Colgate  bought  goods  amounting 
to  $  600.  Dec.  31, 1904,  he  paid  $  300 ;  April  5, 1905,  $  200 ; 
June  1, 1905,  he  settled  the  account.  How  much  did  he  pay 
on  that  date,  if  he  Is  charged  6  %  on  the  purchase  from  its 
date,  and  is  allowed  6  %  interest  on  his  payments  ? 


258 


Chapter  Five. 


4.  John  C.  Kelley  loaned  Chas.  R  Robertson  $  500,  Sept. 
1,  at  6  %.  Payments  of  $  200  each  were  made  Oct.  1  and 
Nov.  1.     How  much  is  due  Dec.  1  ? 


Dr. 


Horace  E.  Dresser 


Or. 


1905. 

1905. 

Feb. 

5 

To   merchan- 
dise, 

840 

00 

Mar. 

9 

By  cash, 

500 

00 

Dec. 

31 

To  interest  to 
date, 

Sept. 
Dec. 

M 

13 
31 
it 

By  cash, 

By  interest  to 

date, 
By  cash, 

200 

00 

5.  Find  the  amount  paid  in  settlement  of  the  foregoing 
account,  Dec.  31,  1905.     Interest  6  %. 

6.  A  merchant's  books  show  the  following  debits :  Feb.  13, 
merchandise,  $725.00;  April  14,  merchandise,  $603.00. 
The  credits  are :  April  5,  cash,  $  600 ;  Aug.  29,  cash,  $  300. 
How  much  is  due  Oct.  5,  interest  6  %  ? 

332.   Oral  Exercises. 

1.  If  I  sell  for  $4.50  a  book  which  cost  me  $3,  what 
per  cent  do  I  gain  ? 

2.  What  is  the  interest  of  $  200,  for  90  days,  at  3%  ? 

3.  One  acre  of  corn  yields  80  bushels,  and  another  acre 
20%  more.     What  does  the  second  acre  yield  ? 

4.  What  will  it  cost  to  fence  a  garden  10  rods  long  and 
6  rods  wide,  at  $  1  a  rod  ? 

5.  In  a  certain  school  40  pupils  are  present  and  10  are 
absent.     What  per  cent  are  absent  ? 

6.  What  is  the  difference  between  a  floor  40  feet  square 
and  two  others  each  20  feet  square  ? 


Miscellaneous.  259 

7.  What  is  the  interest  of  $  12,  for  1  yr.  4  mo.,  at  6%? 

8.  If  2^-  pecks  of  berries  cost  one  dollar,  what  would  3 
quarts  cost  at  the  same  rate  ? 

9.  Bought  5  bushels  nuts  at  a  dollar  a  peck,  and  got  5% 
off  for  cash.     How  much  did  I  pay  for  the  nuts  ? 

333.   Written  Problems. 

1.  Gold  coin  contains  90  per  cent  gold,  9  per  cent  silver, 
1  per  cent  copper.  Find  the  quantity  of  each  metal  in  50 
double-eagles  ($  20),  each  containing  516  grains. 

2.  A,  B,  and  C  buy  a  farm.  A  pays  $8700,  B  pays 
$7200,  C  pays  $4100.  What  per  cent  of  the  purchase 
money  does  each  furnish? 

3.  The  one-cent  pieces  weigh  48  grains.  How  many 
dollars  would  weigh  120  pounds  avoirdupois  (7000  grains  to 
pound) ? 

4.  If  a  person  lends  me  $  250  for  8  months,  for  how 
long  ought  I  to  lend  him  $400  as  an  equivalent? 

5.  Goods  costing  $8  are  sold  at  an  advance  of  20  per 
cent.  The  marked  price  is  $  12,  What  per  cent  reduction 
is  made  on  the  marked  price  ? 

6.  There  are  5  boys  whose  heights  are  4  ft.  9  in.,  5  ft. 
1  in.,  4  ft.  5  in.,  3  ft.  11  in.,  and  4  ft.  4  in.,  respectively. 
What  is  their  average  height? 

7.  In  the  written  number  185.4,  the  number  expressed  by 
the  first  two  (left-hand)  figures  is  how  many  time  the  value 
expressed  by  the  second  two  figures  ? 

8.  Express  decimally,  and  also  as  a  common  fraction,  the 
value  of  each  of  the  following:  115  per  cent;  ^  of  1  per 
cent ;    |~|  of  1  per  cent. 

9.  M  bought  ^  of  a  manufacturing  business  for 
$3517.85,  and  N  bought  -^  of  the  same  business  at  the 
same  rate.     How  much  did  N's  interest  cost  him? 


260  Chapter  Five. 

334.   To  find  Principal,  Kate,  or  Time. 

1.  What  principal  will  produce  $  2.88  interest  in  8  months 
at  4i%  ? 

The  interest  on  $  1  at  4£%  for  8  months  is  $  1  x  gfo  x  T82,  or  $.03. 
Since  $.03  is  produced  by  $  1  (at  the  given  rate  for  the  given  time), 
$2.88  will  require  a  principal  of  as  many  dollars  as  $  .03  is  contained 
times  in  $2.88,  or  96.  Ans.  $96. 

Proof.  $  96  x  gfa  x  A  =  1 2-88- 

2.  What  principal  will  amount  to  $  98.88  in  8  months 
at  4£%  ? 

The  amount  of  $  1  at  4£  %  for  8  months  is  $  1.03.  If  an  amount  of 
$1.03  is  produced  from  a  principal  of  $1,  an  amount  of  $98.88  will 
be  produced  from  a  principal  of  as  many  dollars  as  $  1.03  is  contained 
times  in  $  98.88,  or  96.  Ans.  $  96. 

To  find  the  principal,  divide  the  given  interest  (or  amount) 
by  the  interest  (or  amount)  o/  $  1  at  the  given  rate  for  the  given 
time. 

The  following  is  an  algebraic  method  of  solving  No.  1 : 

(1)  Let  x  represent  the  required  principal. 

(2)  The  interest  will  be    x  X  —  X  — ,  or  ££. 
v  "  200      12        100 

(3)  £^-  =  2.88.     • 
v  J                                         100 

(4)  Clearing  of  fractions,    3  x  =  288. 

(6)  Dividing,  x  =  96.  Ans.  $96. 

The  following  is  an  algebraic  method  of  solving  No.  2 : 

(1)  Let  x  represent  the  required  principal. 

(2)  The  interest  will  be    x  x  —  X  — ,  or  ^ 
K  J  200      12'       100 

(3)  The  amount  will  be     x  +  — ,  or  1^. 
v  '  100  100 

(4)  12^  =  98.88. 
v  J  100 

(6)  Clearing  of  fractions,    103  x  =  9888. 

(6)  Dividing,  x  =  96.  Ans.  $96 


Interest.  261 

3.  At  what  rate  per  cent  will  1 723.60  produce  $  36.18 
interest  in  1  yr.  1  mo.  10  da.  ? 

The  interest  on  $723.60  at  1  %  for  1  yr.  1  mo.  10  da.  is  $723.60 
x  jfa  x  Iff,  or  $8.04.  Since  $8.04  is  produced  by  a  rate  of  1%, 
$36.18  will  require  a  rate  of  as  many  per  cent  as  $8.04  is  contained 
times  in  $  36. 18,  or  4J.  Ans.  4%  %. 

Proof.  $  723.60  x  rfo  X  fff  =  $36.18. 

4.  At  what  rate  per  cent  will  $  723.60  amount  to  f  759.78 
in  1  yr.  1  mo.  10  da.  ? 

Find  the  interest  by  subtracting  the  principal  $723.60  from  the 
amount  $  759.78,  and  proceed  as  in  No.  3. 

To  find  the  rate,  divide  the  given  interest  by  the  interest  at 
1%  on  the  given  principal  for  the  given  time. 

The  following  is  an  algebraic  solution  of  Nos.  3  and  4 : 

(1)  Let  x  represent  the  rate. 

(2)  The  interest  will  be  723.60  x  —  X  — ,  or  8.04  x. 
K  J  100     360 

(3)  8.04  a;  =  36.18. 

(4)  Clearing  of  decimals,      804  x  =  3618. 

(5)  Dividing,  x  =  4£.  Ans.  ±\  %. 

5.  In  what  time  will  $  85.50  produce  $  8.17  interest  at 
4%? 

The  interest  on  $  85.50  at  4  %  for  1  year  is  $  85.50  x  T£7,  or  $  3.42. 
Since  $3.42  is  produced  in  1  year,  $8.17  will  require  as  many  years  as 
$3.42  is  contained  times  in  $8.17,  or  2^. 

Ans.  2T7j  years,  or  2  yr.  4  mo.  20  da. 
Proof.  $  85. 50  x  ^  x  2T\ 

=  $85.50  x^xtf  =  $8.17. 

6.  In  what  time  will  $  85.50  amount  to  $  93.67  at  4%  ? 
'Find  the  interest  by  subtracting  the  principal  $85.50  from  the 

amount  $  93.67,  and  proceed  as  in  No.  5. 

To  find  the  time,  divide  the  given  interest  by  the  interest  for 
i  year  on  the  given  principal  at  the  given  rate. 


262  Chapter  Five. 

335.  The  following  is  an  algebraic  solution  of  Nos.  5 
and  6: 

(1)  Let  x  represent  the  time  in  years. 

(2)  The  interest  will  be  85.50  x  Tfo  x  x,  or  3.42 x. 

(3)  3.42  x  =  8.17. 

(4)  Clearing  of  decimals,   342  x  =  817. 
(6)  Dividing,  x  =  2^. 

Ans.  2T7j  years,  or  2  yr.  4  mo.  20  da. 

336.  The  algebraic  method  consists  (1)  in  representing 
the  unknown  quantity  (principal,  rate,  or  time)  by  x ;  (2)  find- 
ing the  interest,  by  multiplying  principal  by  rate  by  time ; 
(3)  forming  an  equation,  by  making  this  product  equal  to 
given  interest ;  (4)  solving  the  equation. 

337.  Written  Exercises. 
Find  rate,  time,  etc. 

1.  Principal,  $  2000 ;  time,  3  yr. ;  interest,  $  300.   Rate? 

2.  Principal,  $1800;  rate,  4%;  interest,  $144.    Time? 

3.  Time,  8  mo. ;  rate,  4|%  ;  interest,  $  2.88.     Principal? 

4.  Principal,  $  38  ;  time,  2  yr. ;  amount,  $  40.28.    Rate  ? 

5.  Principal,   $140;    rate,   3£%  ;    time,   3  mo.  15  da. 
Interest  ? 

6.  Amount,    $39.60;    rate,   4%;    time,   2    yr.    6    mo. 
Principal  ? 

7.  Amount,    $484.15;     rate,   3J% ;     principal,    $460. 
Time  ?  ■ 

8.  Principal,  $  39.60;  rate,  4%  ;  time,  1  yr.  7  mo.  15  da. 
Amount  ? 

9.  Time,  8  yr. ;  rate,  3%  ;  amount,  $  6200.     Principal  ? 

10.  Principal,  $  7548 ;  time,  3  mo.  5  da. ;  interest,  $  119.51. 
Rate? 

11.  Principal,  $  9000 ;  rate,  4%  ;  interest,  $  632.     Time  ? 

12.  Time,  2  yr.  3  mo.  20  da. ;  rate,  5%  i  amount,  $160.60. 
Principal  ? 


Interest.  263 

13.  Principal,  $  756;  rate,  3\%  ;  time,  3  yr.  4  mo.  20  da. 
Interest  ? 

14.  Principal,  $120;  time,  1  yr.  2  mo.  15  da.;  interest, 
$  4.35.     Rate  ? 

15.  Amount,  $  97.57 ;  rate,  4%  ;  interest,  $  7.57.     Time  ? 

16.  Time,  3  yr.  8  mo.  19  da. ;  rate,  4J%  ;  amount,  $  93.39. 
Principal  ? 

17.  Principal,  $  1848 ;  rate,  3£%;  time,  4  yr.  9  mo.  25  da. 
Amount  ? 

18.  Kate,  5%  ;  time,  4  yr.  6  mo.  23  da. ;  interest,  $  16.43. 
Principal  ? 

338.   Oral  Exercises. 

1.  In  what  time  will  $100  amount  to  $109,  at  6% 
interest  ? 

2.  At  what  rate  will  $  200  produce  $  16  interest  in 
2  years  ? 

3.  What  principal  will  produce  $  12  interest  in  3  years, 
at4%? 

4.  In  what  time  will  $  300,  at  4 % ,  produce  $  29  interest  ? 

5 .  In  what  time  will  $  170  produce  $  1.70  interest,  at  5 %  ? 

6.  In  what  time  will  $  360  produce  $  3.60  interest,  at  4%  ? 

7.  In  what  time  will  $  725  produce  $7.25  interest,  at  6%? 

8.  In  what  time  will  $  45  produce  45  f  interest,  at  4£%? 

9.  In  what  time  will  $  72  produce  $  1.44  interest,  at  6  %  ? 

10.  Find  the  interest  on  $84  for  144  days,  -at  5%. 

11.  Find  the  interest  on  $125,  at   5%,  for  2   months 
12  days. 

12.  At  what  rate  will  $  64  produce  64  i  interest  in  80  days  ? 

13.  At  what  rate  will  $  40  produce  $  1.20  interest  in 
6  months  ? 

14.  A  certain  principal  produces  $  120  interest  at  6%. 
What  would  be  the  interest  if  the  rate  were  4%  ? 


264  Chapter  Five. 

339.   Written  Review  Exercises. 

1.  What  number  increased  by  16%  of  itself  equals  1276? 

2.  A  capitalist  sends  a  commission  merchant  $8670  to 
invest  in  cotton  and  to  include  commission  at  2%.  How 
much  does  the  commission  amount  to  ? 

3.  A  joiner  worked  on  Monday  9  hr.  45  min.,  on  Tuesday 
and  Wednesday  10  hr.  45  min.  each  day,  on  Thursday  and 
Friday  10  hr.  15  min.  each  day,  and  on  Saturday  6  hr.  45  min. 
What  was  the  average  length  of  his  day's  work  ? 

4.  Thirty-two  clerks  are  to  distribute  36,000  letters  on 
a  certain  day.  Half  of  the  clerks  are  experienced  men  and 
half  of  them  new  men.  If  each  experienced  man  does  twice 
as  much  as  a  new  man,  how  many  letters  will  be  distributed 
by  one  of  each  kind  ? 

5.  Sold  my  house  and  farm  of  94£  acres  for  $  12,300. 
Allowing  $  7000  for  the  house,  what  did  I  receive  per  acre 
for  the  land  ? 

6.  A  commission  merchant  receives  $  1071  to  invest  in 
oats  at  30^  per  bushel  and  to  cover  his  commission  at  2% 
for  buying.     How  many  bushels  of  oats  does  he  purchase  ? 

Should  the  commission  merchant  deduct  2  %  of  $  1071,  or  2  %  of  the 
cost  of  the  oats  ? 

7.  What  is  the  total  weight  of  4  hogsheads  of  sugar, 
weighing  respectively  936J,  1025^,  846$,  and  987-^  pounds, 
deducting  tare  at  10  per  cent  ? 

8.  The  product  of  three  factors  is  3289;  two  of  the 
factors  are  23  and  11.     What  is  the  third  factor  ? 

9.  A  man  received  $2.75  per  day,  exclusive  of  Sundays, 
during  1903.  He  paid  $73  for  clothing  for  himself  and 
family,  $  15  per  month  rent,  $  1.10  per  day  for  provisions, 
$  8  per  month  for  fuel  and  light,  and  25  ^  per  day  for  other 
expenses.     How  much  had  he  left  at  the  end  of  the  year  ? 


Bank  Discount.  26$ 

BANK  DISCOUNT. 

340.   Thomas  Tierney,  wishing  to  borrow  three  hundred 

dollars  from  The  Borough  Bank,  draws  up  the  following 

promissory  note : 

Denver,  May  16,  1903. 

Three  months  after  date  I  promise  to  pay  to  the  order 
of  myself  Three  Hundred  Dollars,  value  received,  at  The 
Borough  Bank. 

$300TVo.  Thomas  Tierney. 

As  the  note  now  stands,  it  is  payable  to  Thomas  Tierney. 
He  transfers  by  indorsing  it ;  that  is,  by  writing  his  name 
on  the  back  of  the  note.  The  effect  of  this  indorsement  is 
to  transfer  the  note  to  the  holder,  in  this  case,  the  bank. 
As  a  bank  requires  at  least  a  second  person  as  a  security 
for  the  payment  of  the  loan,  Mr.  Tierney  gets  Herman  A. 
Metz  to  indorse  it  also.  By  this  indorsement,  Mr.  Metz 
agrees  to  pay  the  note  if  Mr.  Tierney  does  not  pay  it  at 
maturity,  August  16. 

The  Borough  Bank  thereupon  pays  over  to  Thomas 
Tierney,  or  places  to  his  credit  on  the  books  of  the  bank, 
the  face  of  the  note  less  the  interest  for  92  days,  $300 
—  $  4.60,  or  $  295.40.  This  interest  taken  in  advance  is 
called  bank  discount.  The  sum  turned  over  to  Thomas 
Tierney  is  called  the  proceeds. 

Face  of  note,  $300.00 

Discount  92  days,  4.60 

Proceeds,  $  295.40 

To  find  the  bank  discount,  compute  the  interest  on  the  face 
of  the  note  from  the  date  of  discount  to  the  date  of  maturity. 
To  find  the  proceeds,  deduct  the  discount  from  the  face. 

Note.  —  The  usage  of  banks  varies  in  different  parts  of  the  country, 
and  the  teacher  should  inform  herself  as  to  the  local  practice. 


i66  Chapter  Five. 

341.  Written  Exercises. 

Find  the  discount  at  6  %  on  the  following : 

1.  A  30-day s  note  for  $75. 

2.  15-day s  note  for  $  183.60. 

3.  60-days  note  for  $275.40. 

4.  20-days  note  for  $  96. 

5.  4-months  note  for  $  336. 
Face  of  note  —  bank  discount  =  proceeds. 

Find  the  proceeds,  at  7%,  on 

6.  A  6-months  note  for  $  180. 

7.  A  3-months  note  for  $  36.90. 

8.  A  24-day s  note  for  $  795.60. 

9.  A  90-day s  note  for  $  180. 
10.   A  72-days  note  for  $  1000. 

342.  Find  the  discount,  at  6%,  on 

11.  A  1-month  note  for  $600,  dated  Feb.  6, 1904.  Due 
March  6,  29  days. 

12.  A  2-months  note  for  $  240,  dated  July  17,  1903. 

13.  A  3-months  note  for  $  360,  dated  April  8,  1904. 

14.  A  4-months  note  for  $  84,  dated  Dec.  24,  1905. 

15.  A  6-months  note  for  $  172.60,  dated  March  4,  1903. 

16.  A  60-days  note  for  $  240,  dated  July  17,  1904. 

17.  A  90-days  note  for  $  360,  dated  April  8,  1903. 

In  each  of  the  preceding  examples,  it  has  been  assumed  that  the 
note  has  been  presents  for  discount  the  day  on  which  it  was  made. 

In  some  of  the  following  examples,  the  notes  are  discounted  at  a 
later  date,  and  the  term  of  discount  is  to  be  ascertained ;  that  is,  the 
time  between  the  date  of  discount  and  that  of  maturity. 

The  term  of  discount  of  a  30-day s  note  dated  May  1,  and  discounted 
May  19,  is  the  time  from  May  19  to  May  31,  12  days. 


Bank  Discount.  267 

343.  In  the  following  examples,  find  (a)  date  of  maturity; 
(b)  term  of  discount ;  (c)  discount ;  (d)  proceeds. 

Note.  — The  pupil  that  works  without  thinking,  frequently  finds  the 

difference  in  time  between  the  two  dates  given  in  the  problem  and  uses 

this  as  the  term  of  discount.     The  time  between  the  dates  given  below 

shows  in  each  case  the  time  the  note  was  not  in  the  possession  of  the 

bank. 

Dated.  Face.  Time.  Discounted.  Kate. 

18.  July  16,1902;   $  87.60;     30  days  ;       August  11,  1902  ;    6% 

This  note  is  due  30  days  after  July  16,  which  is  August  15.  If  the 
bank  discounts  it  August  11,  4  days  before  it  is  payable,  it  deducts 
4  days'  interest,  which  is  6  cents. 

Ansicers  —  Date  of  maturity  Aug.  15,  1902. 
Term  of  discount  4  days. 
Discount  6  cents. 

Proceeds  $87.54. 

19.  Date,  Sept.  9,  1902;  face,  $  124. 18;  time,  4  months; 
discounted,  Nov.  18,  1902;  rate,  *8%. 

20.  Date,  Dec.  5,  1902;  face,  $504.60;  time,  30  days; 
discounted,  Dec.  12,  1902;  rate,  7%. 

21.  Date,  Nov.  14,  1903;  face,  $72.36;  time,  3  months; 
discounted,  Dec.  20,  1903;  rate,  6%. 

22.  Date,  Oct.  30,  1903;  face,  $234;  time,  90  days; 
discounted,  Jan.  5,  1904;  rate,  6%. 

23.  Date,  Jan.  2,  1904;  face,  $95.90;  time,  2  months; 
discounted,  Feb.  13,  1904;  rate,  6%. 

24.  Date,  Aug.  5,  1904;  face,  $164;  time,  60  days; 
discounted,  Aug.  31,  1904;  rate,  8%. 

25.  Date,  Feb.  27,  1904;  face,  $83.20;  time,  100  days; 
discounted,  March  9,  1904;  rate,  6%. 


268  Chapter  Five. 

DISCOUNT  OF  INTEREST-BEARING  NOTES. 

344.   Written  Problems.  ^  „_    -.  -M  «,_. 

Brooklyn,  N.Y.,  Oct.  16,  1904. 

Sixty  days  after  date  I  promise  to  pay  to  the  order  of 
John  Karst,  One  Hundred  Forty-eight  -ffo  Dollars,  value 
received,  with  interest  at  6%. 

$148-/^.  Daniel  Kelly. 

1.  Find  the  proceeds  of  the  above  note  if  discounted 
Dec.  1,  1904,  at  6%. 

At  maturity,  Dec.  15,  1904,  there  is  due  $148.50  with  $1.49  in- 
terest for  sixty  days,  a  total  of  $  149.99.  If  it  is  discounted  Dec.  1, 
14  days  before  maturity,  the  bank  deducts  14  days'  interest  on 
$  149.99,  which  is  35  cents,  and  pays  over  to  John  Karst  the  proceeds, 
$  149.64,  Arts. 

To  find  the  bank  discount  of  an  interest-bearing  note,  com- 
pute the  interest  on  the  amount  due  at  maturity  from  the  time 
of  discount  to  the  date  of  maturity. 

2.  Find  the  proceeds  of,  a  90-day s  note  for  $  175,  bearing 
interest  at  6%,  discounted  33  days  after  date,  at  6%. 

3.  Find  the  proceeds  of  a  60-days  note  for  $  350,  bearing 
interest  at  6%,  discounted  at  6%,  10  days  after  date. 

4.  Find  the  proceeds  of  a  3-months  note  for  $840, 
bearing  interest  at  7%,  discounted  at  bank  47  days  before 
maturity,  at  8%. 

5.  A  4-months  note  for  $720,  dated  March  17,  1905, 
bearing  interest  at  6%,  is  discounted  at  7%,  May  10.  What 
are  the  proceeds  ? 

6.  The  following  note  was  discounted  at  6%,  Sept.  19, 
1904.     Find  the  proceeds. 

Milwaukee,  Wis.,  June  30,  1904. 

Four  months  after  date  I  promise  to  pay  Thomas  Cacciola, 
or  order,  Five  Hundred  Dollars,  value  received,  with  interest 
at  6  per  cent. 

$500^.  George  H.  Greene. 


Bank  Discount.  269 

345.  To  find  the  face  of  note,  rate  of  discount,  or  term. 

1.  The  discount  at  6%  on  a  note  having  84  days  to  run, 
is  $  10.50.     Find  the  face  of  the  note. 

The  discount  on  $1  for  84  days  at  6%  =  $1  x  y-Jfo  X  flfo  or  $.014. 
If  $.014  is  the  discount  on  a  note  for  $  1,  $10.50  will  be  the  discount 
on  a  note  for  as  many  dollars  as  $.014  is  contained  times  in  $10.50,  or 
$750.     Ans.  $750. 

2.  The  proceeds  of  a  note  having  84  days  to  run,  dis- 
counted at  6%,  are  $739.50.     Find  the  face  of  the  note. 

The  discount  on  $1  for  84  days  at  6%  is  $.014 ;  the  proceeds  are 
$1  -$.014,  or  $.986.  If  $.986  are  the  proceeds  of  a  note  for  $1, 
$739.50  will  be  the  proceeds  of  a  note  for  as  many  dollars  as  $.986  is 
contained  times  in  $739.50,  or  $750.    Ans.  $750. 

To  find  the  face  of  a  note,  divide  the  given  discount  {or  pro- 
ceeds) by  the  discount  (or  proceeds)  of  $1  for  the  given  term 
at  the  given  rate. 

3.  The  discount  on  a  note  for  $  750  having  84  days  to  run 
is  $  10.50.     What  is  the  rate  of  discount  ? 

The  discount  on  $750  at  1%  for  84  days  is  $750  x  TJ7x  /fa,  or  $1.75. 
If  $1.75  is  produced  by  a  rate  of  1%,  $10.50  will  be  produced  by  a  rate 
of  as  many  per  cent  as  $1.75  is  contained  times  in  $10.50,  or  6%. 

To  find  the  rate  of  discount,  divide  the  given  discount  by  the 
discount  at  1  %  on  the  given  sum  for  the  given  term. 

4.  The  discount  at  6%  on  a  note  for  $750  is  $10.50. 
How  many  days  has  the  note  to  run? 

The  discount  on  $750  at  6%  for  1  day  is  $750x^X3-^,  or  $.125. 
If  $.125  is  the  discount  for  1  day,  $10.50  will  be  the  discount  for  as 
many  days  as  $.125  is  contained  times  in  $10.50,  or  84  days. 

To  find  the  term  of  discount,  divide  the  given  discount  by  the 
discount  for  1  day  on  the  given  sum  at  the  given  rate. 

346.  Note. — To  solve  by  the  algebraic  method,  use  x  to  represent 
the  unknown  quantity. 


270  Chapter  Five. 

347.   Written  Exercises. 

1.  Three-months  note;  face,  $108;  rate,  6%.  Find 
proceeds. 

2.  90-days  note ;  face,  $  360 ;  discount,  $  6.30.   Find  rate. 

3.  Proceeds,  $717.60 ;  rate,  5%  ;  face,  $  720.    Find  term. 

4.  Discount,  $  J  1.20 ;  rate,  7%  ;  term,  48  days.  Find  face. 

5.  15-days  note ;  face,  $  1560 ;  rate,  6%.     Find  discount. 

6.  Term,  20  days;  face,  $158.40;  proceeds,  $157.96. 
Find  rate. 

7.  Eate,  7%;  discount,  $2.10;  face,  $150.     Find  term. 

8.  Two-months  note ;  discount,  $  14.70 ;  rate,  7%.  Find 
face. 

9.  For  what  amount  must  a  60-day s  note  be  drawn  so 
that  the  proceeds  will  be  $  300  when  the  rate  of  discount  is 
8  per  cent  ? 

10.  A  note  for  $  120  was  discounted  at  a  bank  March  15, 
1905.  What  is  the  date  of  the  maturity  of  the  note,  the 
proceeds  being  $119.52  and  the  rate  of  discount  6  per  cent  ? 

11.  Find  the  proceeds  of  a  6-months  note  for  $875  drawn 
Jan.  2, 1906,  and  discounted  at  6  per  cent  35  days  after  that 
date. 

12.  A  merchant  bought  300  barrels  of  flour  at  $4.75  per 
barrel,  cash,  and  sold  it  for  $5  per  barrel,  taking  in  payment 
a  60-days  note  for  the  amount.  If  he  has  the  note  dis- 
counted immediately  at  a  bank,  at  7  per  cent,  what  does 
he  gain  by  the  transaction  ? 

13.  What  will  be  the  face  of  a  30-day s  note,  the  proceeds 
of  which  when  discounted  at  a  bank  at  6%  will  pay  for 
3000  bushels  corn  at  49|^  per  bushel  ? 

14.  The  proceeds  of  a  note  for  $1200,  due  March  15, 
1904,  and  discounted  at  6%,  were  $1184.80.  When  was  it 
discounted  ? 


Interest.  271 

INTEREST  BY  ALIQUOT  PARTS. 

348.   Written  Exercises. 

1.  Find  the  interest  on  $387.45,  for  2  yr.  8  mo.  18  da., 

at7%-  $387.45  x  .07. 

$27.1215  interest  for  1  yr. 

27.1215  interest  for  1  yr. 

6  mo.  =  \  yr.  13.5607  interest  for  6  mo. 

2  mo.  =  \  (of  6  mo.)        4.5202  interest  for  2  mo. 
15  da.  =  |  (of  2  mo.)         1.1301  interest  for  15  da. 

3  da.  =  $  (of  15  da.)         .2260  interest  for  3  da. 

Ans.  $73.68      interest  for  2  yr.  8  mo.  18  da. 

2.  Find  the  interest  on  $432.90,  at  6%,  for  1  yr.  7  mo. 

12  da"  $432.90  x  .06. 

interest  for  1  yr. 
6  mo.  =  \  yr.  interest  for  6  mo. 

1  mo.  =  \  (of  6  mo. )         interest  for  1  mo. 
10  da.  =  \  (of  1  mo.)         interest  for  10  da. 

2  da.  =  \  (of  10  da.)  interest  for  2  da. 

interest  for  1  yr.  7  mo.  12  da. 

3.  Find  the  amountf  of  $874.16,  at  5%,  for  1  yr.  9  mo. 

4  da*  $874.16    principal. 

5  %  =  ^           •  43.708  interest  for  1  yr. 

6  mo.  =  \  yr.  interest  for  6  mo. 

3  mo.  =  \  (of  6  mo.)  interest  for  3  mo. 
3  da.  =  ^  (of  3  mo.)  interest  for  3  da. 
1  da.   —\  (of  3  da.)        interest  for  1  da. 

amount  for  1  yr.  9  mo.  4  da. 

4.  What  is  the  amount  of  $95.72,  for  3  yr.  6  mo.  20  da., 

at5%?  $95.72    principal. 

10  %  =  ^  9. 572  interest  for  2  yr. 

1  yr.   =  \  (of  2  yr.)  4.786  interest  for  1  yr. 

6  mo.  =  \  yr.  interest  for  6  mo. 

20  da.  =  ?  of  6  mo.  ___^___  interest  for  20  da. 

amount  for  3  yr.  6  mo.  20  da. 


0.J2  Chapter  Five. 

5.  Interest  of  $1806.45,  at  4%,  for  1  yr.  7  ino.  25  da, 

1  yr.,  6  mo.,  1  mo.,  15  da.,  5  da.,  5  da. 

6.  Interest  for  10  mo.  29  da.,  at  4%,  on  $380.40. 

$380.40  x  .04. 
$15.2160  interest  for  1  yr. 
1  mo.  =  ^  yr.  interest  for  1  mo.  )  deduct  from  interest 

1  da.  =  ^  mo.  interest  for  1  da.  )      for  1  yr. 

interest  for  10  mo.  29  da. 

7.  Amount,  at  6%,  of  $125.73,  for  2  yr.  10  mo..  4  da. 

8.  Interest  on  $84.66,  at  7%,  for  1  yr.  4  mo.  12  da. 

9.  Interest,  at  5%,  for  4  yr.  2  mo.  7  da.,  on  $250. 

10.  Amount  of  $1000,  at  6%,  for  33  days. 

349.   When  the  time  is  less  than  a  year,  the  following 
facts  should  be  remembered: 

6%   for  a  year  is  1  per  cent  for  60  days. 
5%   for  a  year  is  1  per  cent  for  72  days. 
4£%  for  a  year  is  1  per  cent  lor  ?  days. 
4%   for  a  year  is  1  per  cent  for  ?  days. 

11.  Find  the  interest  for  81  days,  at  5%,  on  $876.40. 
Since  5%  for  a  year  is  1%  for  72  days,  we  have  :  — 

72  days'  interest  is  1%  of  principal,  or  $8,764 
9  days'  interest  is  $  of  72  days,  or  1.095 

$9.86     interest  for  81  days. 

12.  Amount  of  $954,  at  4%,  for  4  mo.  10  da. 

Principal  $954.00 
3  months'  interest  =  1%  9.54 
1  mo.  =  $  (of  3  mo.)  3J8 

10  da.  =  |(of  1  mo.)         

amount  for  4  mo.  10  da. 


Interest.  273 

13.  Interest  of  $1874,  at  4£%,  for  93  da. 

80  days  =  1% 
10  days 

2  days 

1  day 

14.  Interest  of  $753.20,  at  5%,  for  158  days. 

72  da.,  72  da.,  12  da.,  2  da. 

15.  Amount  of  $1234.50,  for  193  days,  at  6%. 

60  da.,  120  da.,  12  da.,  1  da. 

16.  Find  the  proceeds  of  a  90-day s  note,  for  $873.60, 

at  6^*  Face    $873.60 

60  da.         8.7361  ^  J 

y  Deduct. 
30  da.         4. 368  J 

$860.50  proceeds. 

17.  Find  the  discount  on  a  3-months  note,  for  $1596, 

at  6%. 

18.  What  are  the  proceeds  of  a  6-months  note,  for  $  785, 
discounted  at  6%. 

19.  Find  the  interest  on  $484.40,  for  1  yr.  3  mo.  17  da., 

at  7%. 

20.  Find  the  amount  of  $  683,  for  3  yr.  4  mo.  11  da.,  at 

350.    N.B.  —  Do  not  use  unnecessary  figures. 

21.  Principal,  $360;  5%  ;  3  yr.  7  mo.  18  da.     Interest? 

22.  Principal,  $  613 ;  4|%  ;  157  da.     Amount  ? 

23.  Principal,  $1774;  3|%  ;  17  mo.  23  da.     Interest? 

24.  Principal,  $  875;  6%  ;  2  yr.  3  mo.  1  da.     Amount? 

25.  Principal,  $976;  7%;  325  da.     Interest? 


274  Chapter  Five. 

351.  By  the  time  of  a  note  is  meant  the  number  of  days,  etc.,  for 
which  it  is  drawn.  In  these  four  examples  the  note  is  discounted  the 
day  it  is  made. 

26.  Face  of  note,  $  254 ;  time,  30  days ;  7%.     Proceeds  ? 

27.  Face  of  note,  $515;  time,  6  months;  5%.  Dis- 
count ? 

28.  Face  of  note,  $493;  time,  60  days;  8%.  Proceeds? 

29.  Face  of  note,  $717;  time,  15  days;  6^%.  Discount? 

352.  Find  the  exact  number  of  days.    Take  360  days  to  year. 

30.  Principal,  $1836.50;  6%;  Jan.  2  to  Dec.  1.    Amount? 

31.  Principal,  $1295.70;  7%;  March  8  to  April  9. 
Interest  ? 

32.  Principal,  $  765.90;  4% ;  Oct.  1  to  Dec.  17.     Interest? 

33.  Principal,  $275.84;  51%;  May  9  to  July  3.    Amount? 

353.  By  the  term  of  a  note  is  meant  the  number  of  days  it  has  to 
run  after  it  has  been  discounted. 

34.  Face  of  note,  $100;  term,  60  days;  7%.  Discount? 

35.  Face  of  note,  $ 200 ;  term,  90  days;  6£%.  Proceeds? 

36.  Face  of  note,  $300;  term,  24  days;  5J%.  Discount? 

37.  Face  of  note,  $400;  term,  117  days;  8%.  Proceeds? 

354.  In  examples  38-41,  inclusive,  find  the  time  by  compound  sub- 
traction. 

38.  Principal,  $25.83;  6%;  Jan.  14,  1902,  to  Sept.  5, 
1904.     Interest  ? 

39.  Principal,  $47.96;    5%;    Feb.  6,  1903,  to  Aug.  1, 

1906.  Amount  ? 

40.  Principal,  $85.30;  7%  ;  March  25,  1904,  to  Jan.  13, 

1907.  Interest  ? 

41.  Principal,  $75;  4%;  April  15,  1900,  to  Feb.  6, 
1907.     Amount? 


Review.  275 

REVIEW. 
355.   Oral  Problems. 

1.  Out  of  500  pupils,  50  are  absent.  What  is  the  per 
cent  of  attendance  ? 

2.  A  can  do  a  piece  of  work  in  4  days ;  B  can  do  it  in 

4  days.     In  what  time  can  A  and  B  do  it,  if  they  work 
together  ? 

3.  What  is  the  interest  of  $  1500,  for  60  days,  at  6%  ? 

4.  In  a  certain  class  \  of  the  pupils  are  under  10  years, 
-J-  of  them  are  between  10  and  12,  and  the  rest  are  over  12. 
What  per  cent  are  over  12  years  ? 

5.  If  a  bushel  of  English  walnuts  costs  $  1.60,  what  will 
6  quarts  cost  ? 

6.  A  man  put  5  gal.  2  qt.  of  syrup  into  bottles  holding 
2  quarts  each.     How  many  bottles  did  it  require  ? 

7.  If  I  of  a  yard  of  cloth  costs  £  of  a  dollar,  what  will  J 
of  a  yard  cost  ? 

8.  If  9  pounds  of  sugar  cost  48  ^,  what  will  12  pounds 
cost? 

9.  What  is  the  difference  between  f  of  8§  and  f  of  4 J  ? 

10.  How  many  eggs,  at  the  rate  of  15  for  25  cents,  can  be 
bought  for  60^? 

11.  A  merchant's  receipts  are  $1200;  his  gain  is  20  per 
cent.     What  part  of  his  receipts  is  profit  ? 

12.  If  3  men  earn  $72  in  8  days,  how  many  dollars  will 

5  men  earn  in  11  days  ? 

13.  If  a  dealer  loses  25%  by  selling  a  horse  for  $  225, 
what  per  cent  would  he  gain  or  lose  by  selling  the  horse  for 
$325? 

14.  If  A  can  do  a  piece  of  work  in  2  days,  B  in  3  days, 
and  C  in  4  days,  in  what  time  can  they  do  it,  working  to- 
gether ? 


276  Chapter  Five. 

356.    Written  Problems. 

1.  A  man  sold  18  barrels  sugar,  each  containing  306 
pounds ;  21  barrels,  each  containing;  297  pounds ;  5  barrels, 
each  containing  291  pounds.  What  is  the  average  weight 
per  barrel  ? 

2.  Three  men  engage  in  a  business  venture.  One  fur- 
nishes if  3000,  another  furnishes  $  5000,  a  third  furnishes 
$  4000.  They  gain  $  1800.  What  is  each  one's  share  of 
the  profit  ? 

What  part  of  the  money  did  the  first  furnish  ?  What  part  of  the 
profit  should  he  receive  ? 

3.  Three  ounces  is  what  per  cent  of  5  pounds  ? 

4.  What  is  the  product  of  \  of  f  of  15|.  State  the  re- 
sult in  decimals. 

5.  What  is  87£%  of  $896?  $896  is  87£  %  of  what 
sum? 

6.  What  number  is  that  which,  diminished  by  2 J,  will 
leave  2^- 

7.  How  long  will  200  pounds  flour  last  18  persons  if 
each  person  is  allowed  If  pounds  per  day  ? 

8.  If  f  of  I  of  a  ship  cost  $  84,000,  what  is  £  of  it  worth  ? 

9.  The  dividend  was  $  4689.036,  the  quotient  .027,  what 
was  the  divisor? 

10.  Harry  Hedge  earns  $  12  a  week.  He  pays  $  4.25  for 
board,  $0,625  for  car  fare,  $0,375  for  library  fees,  and 
$  4.875  for  other  expenses.  In  how  many  weeks  would  he 
save  $  97.50. 

11.  For  how  long  must  $450  be  at  interest,  at  five  per 
cent  per  annum,  to  amount  to  $  481.62  ? 

12.  Divide  320  acres  of  land  among  A,  B,  and  C,  so  that 
A  shall  have  15  acres  more  than  B,  and  C  shall  have  27 
acres  more  than  B. 


Denominate  Numbers.  277 

DENOMINATE  NUMBERS. 

357.  Inductive  Exercises. 

1.  Change  1  yd.  1  ft.  to  inches. 

2.  Change  1  yd.  1  ft.  1  in.  to  inches. 

3.  Change  49  inches  to  yards,  feet,  etc 

4.  Change  49  pints  to  gallons,  etc. 

5.  Add  4  lb.  8  oz.  and  4  lb.  8  oz. 

6.  From  9  pounds  take  4  lb.  8  oz. 

7.  Multiply  4  lb.  8  oz.  by  2. 

8.  Divide  9  pounds  by  2. 

9.  Divide  9  pounds  by  4  lb.  8  oz. 

10.  How  many  inches  in  J  yd.  ? 

11.  How  many  feet  and  inches  in  .75  yd.  ? 

12.  75  per  cent  of  a  yard  =  ? 

13.  What  fraction  of  a  yard  is  27  inches  ? 

14.  Change  2  ft.  3  in.  to  the  decimal  of  a  yard. 

15.  1  ft.  6  in.  is  what  per  cent  of  2  feet? 

16.  Multiply  9  yd.  18  in.  by  7. 

17.  From  18  lb.  6  oz.  take  9  lb.  12  oz. 

358.  Troy  Weight. 

24  grains  (gr.)    =  1  pennny  weight  (pwt.) 
20  pennyweight  =  1  ounce  (oz.) 
12  ounces  =  1  pound  (lb. ) 

Troy  weight  is  used  in  weighing  gold,  silver,  precious  stones,  etc. 

359.  English  Money. 

12  pence  (<f.)      =1  shilling  (s.) 
20  shillings  =  1  pound  (£) 

A  farthing  is  a  quarter  of  a  penny. 


278  Chapter  Five. 

REDUCTION  DESCENDING. 

360.    Change  43  yd.  2  in.  to  inches. 

Write  43  yd.  0  ft.  2  in.,  inserting  the  3  ft.       12  in. 


missing  denomination,  feet.  Above  0  ft.  43  yd. '  0  f t.  2  in. 
write  the  number  of  feet  in  a  yard,  and  T20  ft.  1550  in 

above  the  2  in.  the  number  of  inches  in  a  A       1  ~~n  . 

foot.    Since  there  are  3  feet  in  a  yard,  in 

43  yards  there  are  43  times  3  feet,  or  129  feet.  This  is  written  in  the 
column  of  feet.  Since  there  are  12  inches  in  a  foot,  in*  129  feet  there  are 
129  times  12  inches,  or  1548  inches.  Add  2  inches,  making  1550  inches, 
which  is  written  in  the  column  of  inches,  and  cancel  129  feet. 

In  working  this  example,  3  and  12  are  used  as  the  multipliers  instead 
of  43  and  129.  At  the  time  the  9  of  129  is  multiplied  by  12,  the  2  is 
added  in,  the  pupil  saying  12  nines  are  108,  and  2  are  110,  writing  the 
0  ;  12  twos  are  24,  and  1 1  are  35,  writing  the  5  ;  etc. 

361.   Written  Exercises. 
Change : 

1.  4  yards  2  feet  8  inches  to  inches. 

2.  2  miles  46  rods  3  yards  to  yards. 

3.  3  pecks  5  quarts  1  pint  to  pints. 

4.  6  bushels  3  pecks  6  quarts  to  quarts. 

5.  2  gallons  3  quarts  1  pint  to  pints. 

6.  7  gallons  1  quart  1  pint  to  pints. 

7.  4  ounces  12  pennyweights  3  grains  to  grains, 

8.  2  pounds  16  pennyweights  14  grains  to  grains. 

9.  6  pounds  9  shillings  7  pence  to  pence. 

10.  8  pounds  18  shillings  4  pence  to  pence. 

11.  3  wk.  4  da.  13  hr.  to  hours. 

12.  I  of  a  week  to  hours. 

13.  -J  of  a  mile  to  yards. 

14.  .25  of  a  rod  to  inches. 


Denominate  Numbers.  279 

REDUCTION  ASCENDING. 

362.  Change  1550  inches  to  yards,  feet,  etc. 

Above  1550  in.  write  12  in.,  the  num-  3  ft.       12  in. 

ber  in  a  foot.     Dividing  1550  inches  by  729  ft.  1550  in. 

12  inches  we  obtain  the  quotient  129,  the  43  y^       0  ft.         2  in. 
number  of  feet,  and  2  inches  remainder.  A        .0     ,    0  . 

w  *     *u  >  a      •     a        1  m  ^ns-  43  yd-  2  in. 

Write  the  remainder  in  the  column  of  ■* 

inches  and  129  ft.  to  the  left  of  1550  in.     Reduce  129  ft.  to  yards, 

writing  the  result,  43  yd.,  as  shown  above,  and  cancelling  129  ft. 

363.  "Written  Exercises. 
Change : 

1.  4530  feet  to  rods,  yards,  etc. 

2.  6324  yards  to  miles,  rods,  etc. 

3.  244  pints  to  bushels,  pecks,  etc. 

4.  467  quarts  to  bushels,  pecks,  etc. 

5.  923  pints  to  gallons,  quarts,  etc.    • 

6.  785  pints  to  gallons,  quarts,  etc. 

7.  543  pennyweights  to  pounds,  etc. 

8.  175  grains  to  pennyweights,  etc. 

9.  625  pence  to  pounds,  shillings,  etc. 

10.  836  shillings  to  pounds,  etc. 

11.  8423  min.  to  days,  hours,  etc. 

12.  2348  inches  to  yards,  etc. 

ADDITION  OF  DENOMINATE  NUMBERS. 

8  in.  +  10  in.  +  5  in.  =  23  in.  =  1  f t 
11  in.     Write  11  in.  and  carry  1  ft. 

1  f t.  +  1  ft.  +  2  f t.  =  4  f t.  =  1  yd.  1  ft. 
Write  1  ft.  and  carry  1  yd. 
16  yd.     1ft.     11  in.  Ana.       1  yd. +4  yd. +  11  yd.  =  16  yd.    Write 

16  yd. 


364. 

Find  the  sun 

Hyd. 

2  ft. 

8  in. 

1ft. 

10  in. 

4  yd. 

Oft. 

5  in. 

28o  Chapter  Five. 

365.   Written  Exercises. 

Find  sums : 

1.   8  mi.      44  rd.     3  yd.  4.   243  gal.     2  qt.     1  pt. 

6  mi.     298  rd.     4  yd.  168  gal.     3  qt.    1  pt. 

67  rd.     1  yd.  1  qt.     1  pt. 


27  rd. 

3  yd. 

2  ft. 

5.  4  1b. 

10  oz. 

14  pwt. 

3rd. 

2  yd. 

1ft. 

3  1b. 

9oz. 

16  pwt. 

78  rd. 

4  yd. 

2  ft. 

lib. 

11  oz. 

7  pwt. 

8bu. 

3pk. 

5qt. 

6.  8  oz. 

9  pwt. 

21  gr. 

16  bu. 

2pk. 

3qt. 

3oz. 

11  pwt. 

6gr. 

4bu. 

3pk. 

7qt. 

17  pwt. 

23  gr. 

•3. 


SUBTRACTION  OF  DENOMINATE  NUMBERS. 

From  35  yd.    1  ft.   4  in.  Since  8  in.  is  greater 

Take  19  yd.    2  ft.   8  in.  than  4  in->  w©  must  use 

15  yd.    1  ft.   8  in.  Ans.     l  ft'  4  ,n-  or  16  in"  M 

the  minuend.    16  in.  — 

8  in.  =  8  in.    As  the  minuend  now  contains  0  ft.,  1  yd.  is  taken  from 

35  yd.    Changing  the  yard  to  3  ft.,  and  deducting  2  ft.,  leaves  1  ft, 

U  yd.  -  19  yd.  =  15  yd. 

867.   "Written  Exercises. 
Find  differences : 


1. 

183  rd.  4  yd. 
68  rd.  5  yd. 

1ft. 
2  ft. 

2. 

91  mi.  83  rd. 
26  mi.  122  rd. 

2  yd. 
4  yd. 

3, 

3  pk.  1  qt. 
1  pk.  4  qt.  1 

pt. 

4.   29  gal.     2  qt. 

28  gal.    3  qt.     1  pt. 


5. 


6. 


8  1b. 
6  1b. 

3  oz.  8  pwt. 
8  oz.  10  pwt. 

£24 
£3 

6s.  3d. 
9s.  Sd. 

Denominate  Numbers.  281 

MULTIPLICATION  OF  DENOMINATE  NUMBERS. 

368.  Multiply    34  yd.  2  ft.  9  in.  by  7. 

244  yd.  1  ft.  3  in. 

7  times  9  in.  =  63  in.  =  5  f t.  3  in.  Write  3  in.  7  times  2  ft.  = 
14  ft.  Carry  5  ft.,  making  19  ft.,  or  6  yd.  1  ft.  Write  1  ft.  Multiply 
34  yd.  by  7,  adding  in  6  yd.  when  the  4  is  multiplied. 

369.  Written  Exercises. 
Find  products : 

1.  6  mi.  24  rd.  4  yd.  by  9.  6.  3  gal.  1  qt.  1  pt.  by  32. 

2.  36  rd.  4  yd.  2  ft.  by  12.  7.  8  lb.  4  oz.  12  pwt.  by  10. 

3.  24  bu.  3  pk.  6  qt.  by  14.  8.  16  oz.  12  pwt.  20  gr.  by  4. 

4.  2  pk.  3  qt.  1  pt.  by  36.  9.  £4  12s.  6d.  by  20. 

5.  11  gal.  2  qt.  1  pt.  by  8.  10.  £28  16s.  9d.  by  7. 

DIVISION  OF  DENOMINATE  NUMBERS. 

370.  Divide    244  yd.  1  ft.  3  in.  by  7. 

34  yd.  2  ft.  9  in.  Ans. 

The  quotient  of  244  yd.  divided  by  7  is  34  yd.,  with  a  remainder  of 
6  yd.  Reducing  6  yd.  to  ft.  and  adding  in  1  ft.,  the  dividend  is  19  ft. 
19  ft.  4-  7  =  2  ft.  with  5  ft.  remainder.  5  ft.  3  in.  =  63  in.  63  in.  -*-  ? 
=  9  in. 

371.  "Written  Exercises. 
Find  quotients : 

1.  44  mi.  124  rd.  2  yd.  by  8. 

2.  14  yd.  1  ft.  9  in.  by  21. 

3.  37  bu.  1  pk.  2  qt.  by  6. 

4.  12  bu.  3  pk.  6  qt.  by  18. 

5.  7  gal.  3  qt.  1  pt.  by  3. 


282  Chapter  Five. 

6.  3  gal.  1  pt.  by  5. 

7.  28  lb.  10  oz.  16  pwt.  by  24. 

8.  4  oz.  10  pwt.  3  gr.  by  9. 

9.  £  24  l»7s.  4d.  by  16. 
10.  £  3  7s.  6d.  by  10. 

372.  Divide  244  yd.  1  ft.  3  in.  by  34  yd.  2  ft.  9  in. 

In  dividing  one  concrete  number  by  another  concrete  number,  the 
divisor  and  the  dividend  must  be  of  the  same  denomination.  Thus, 
to  divide  $  2  by  25^,  we  change  the  dividend  to  cents,  200  cents  -f-  25 
cents,  or  the  divisor  to  dollars,  $  2  -f-  $  \.  •  The  quotient  is  8,  an  abstract 
number  ;  that  is,  25  cents  is  contained  in  200  cents  8  times. 

244  yd.  1  ft.  3  in.  =  8799  in,  ;  34  yd.  2  ft.  9  in.  =  1257  in.  8799  in. 
-T- 1257  in.  =  7,  Ans. 

The  result  would  be  the  same  if  we  divided  733£  ft.  by  104£  ft.,  or 
244&  yd.  by  34H  yd. 

373.  Written  Exercises. 
Find  quotients : 

1.  4  mi.  36  rd.  1  yd.  by  6  rd.  3  yd. 

2.  88  rd.  2  yd.  2  ft.  by  8  rd.  4  yd.  2  ft 

3.  21  bu.  2  pk.  4  qt.  by  1  bu.  3  pk.  4  qt 

4.  15  bu.  1  pk.  by  3  pk.  6  qt.  1  pt. 

5.  60  gal.  1  pt.  by  4  gal.  2  qt.  1  pt. 

6.  16  gal.  3  qt.  1  pt.  by  2  qt.  1  pt. 

7.  17  lb.  11  oz.  10  pwt.  by  8  lb.  11  oz.  15  pwt. 

8.  1  lb.  2  oz.  18  pwt.  by  3  oz.  14  pwt.  12  gr. 

9.  £24  16s.  8d  by  £18  12s.  6<f. 
10.  £2  14s.  3d.  by  £  8  2s.  90. 


Denominate  Numbers.  283 

374.  Oral  Problems. 

1.  What  will  be  the  cost  of  3  lb.  7  oz.  of  tea,  at  64^  per 
pound? 

2.  How  many  feet  in  2\  rods  ? 

3.  At  37-^  per  peck,  what  shall  I  receive  for  4  bushels 
of  potatoes  ? 

4.  What  will  be  the  cost  of  a  ton  of  hay  at  97 \$  per 
cwt.  ? 

5.  If  slate  pencils  cost  2  mills  each,  how  many  can  be 
bought  for  $4? 

6.  At  $5.00  per  ton,  how  many  pounds  of  coal  can  be 
bought  for  Iff 

7.  Find  the  cost  of  3  T.  480  lb.  coal  at  $5  per  ton. 

8.  At  $5  per  ton,  how  many  tons  and  pounds  of  coal 
can  I  buy  for  $10.80? 

9.  Find  the  cost  of  4  yd.  1  ft.  of  ribbon,  when  2  yd.  2  ft. 
cost  40  cents. 

10.  In  2|  pecks,  how  many  quarts  ? 

11.  How  many  hours  in  |  of  a  day  ? 

12.  1.25  pecks  are  how  many  quarts  ? 

13.  At  $12  per  ounce,  what  is  f  of   a  pound  of  gold 
worth  ? 

14.  How  many  feet  in  a  quarter  of  a  mile  ? 

15.  How  many  tablespoons,  each  weighing  2  ounces,  can 
be  made  from  2  lb.  10  oz.  of  silver  ? 

375.  Written  Problems. 

1.  What  will  be  the  cost  of  150  yards  silk  at  3/6  per 
yard  ?  3/5  _  3s  ^  t  rea(i  three  an(j  sixpence. 

2.  If  £  1  =  $  4.8665,  what  will  be  the  cost  in  U.  S.  money 
of  75  books  at  18  pence  each  ? 


284  Chapter  Five. 

3.  A  merchant  sells  37  coats  at  £  3  5s.  each,  less  10%. 
What  is  the  amount  of  his  bill  in  English  money  ? 

4.  Find  25%  of  £  183  14s.  8d. 

5.  A  silver  dollar  weighs  412J  grains.  How  many  ounces 
of  pure  silver  are  there  in  1000  silver  dollars  if  the  coin  is 
y9^  pure  silver  ? 

6.  The  wheels  of  an  engine  being  16  ft.  8  in.  in  circum- 
ference, and  the  number  of  revolutions  150  per  minute,  how 
far  does  it  go  in  an  hour  ?     Give  answer  in  miles  and  rods. 

7.  What  fractional  part  of  30  rd.  5  yd.  1  ft.  is  8  rd.  4  yd. 
2ft.? 

8.  What  decimal  part  of  a  mile  is  39.27  yd.  ? 

9.  3  bu.  1  pk.  5  qt.  is  what  per  cent  of  20  bu.  1  pk.  6  qt.  ? 

10.  If  a  letter-carrier  in  delivering  letters  takes  47,520 
•steps  in  a  day,  each  step  averaging  20  inches,  how  many 
miles  does  he  walk  ? 

11.  43  gal.  3  qt.  1  pt.  alcohol  are  sold  for  $  70.20.  What 
is  the  price  per  gallon  ? 

12.  After  taking  out  15%  of  the  grain  in  a  bin,  there  re- 
mained 40  bu.  3  pk.  5  qt  How  many  bushels  were  there  at 
first? 

13.  A  merchant  bought  51  tons  17  cwt.  3  qr.  25  lb.  of 
wool,  and  sold  27  tons  4  cwt.  2  qr.  27  lb.  Of  the  remainder, 
one-half  was  lost  by  fire.     How  much  had  he  left  ? 

28  lb.  =  1  quarter  ;  4  quarters  =  1  cwt. 

14.  An  invoice  of  wool  weighs  32  tons  17  cwt.  2  qr.  11  lb. 
State  the  value  in  £  s.  d.f  at  lOd.  sterling  per  pound. 

1  ton  =  2240  lb. 

15.  How  many  minutes  in  February,  1904  ? 


Denominate  Numbers.  285 

16.  If  a  locomotive  runs  25  mi.  48  rd.  in  50  minutes, 
how  far  will  it  run  in  12  hours  ? 

Give  answer  in  miles  and  decimals  of  a  mile. 

17.  I  wish  to  put  111  bu.  2  pk.  4  qt.  of  grain  into  47 
bags.     What  quantity  must  each  contain  ? 

18.  If  a  river  current  carries  a  raft  of  lumber  at  the  rate 
of  4  miles  180  rods  per  hour,  how  long  will  it  take  the  raft 
to  float  365  miles? 

19.  Bought  28,500  pounds  of  hay  at  $  12^  a  ton,  and  sold 
it  at  $  0.87^  per  hundredweight.     What  was  the  gain  ? 

376.  1  pound  Troy  =  5760  grains. 

1  pound  Apothecaries'  =  5760  grains. 

1  pound  Avoirdupois    =  7000  grains. 
How  many  grains  in  a  Troy  ounce  ?    In  an  Avoirdupois  ounce  ? 

1.  Find  the  value  of  a  dozen  silver  spoons,  each  weigh- 
ing 3  oz.  5  pwt,  at  $  1.20  per  oz. 

2.  A  gold  chain  weighs  384  grains.  What  is  its  cost  at 
$1.15  per  pwt? 

3.  Add  4  lb.  6  oz.  18  gr.,  5  oz.  9  pwt.,  3  lb.  20  gr.,  and 
9  lb.  11  oz.  15  pwt.  5  gr. 

4.  How  many  spoons,  each  weighing  2  oz.  18  pwt.,  can 
be  made  from  5  lb.  9  oz.  12  pwt.  silver  ? 

5 .  What  fraction  of  a  pound  Avoirdupois  is  a  pound  Troy? 
What  per  cent  of  an  ounce  Avoirdupois  is  a  Troy  ounce  ? 

6.  What  is  the  value,  at  $  1.60  per  oz.  Troy,  of  a  silver 
pitcher  weighing  4  lb.  8  oz.  Avoirdupois  ? 

7.  At  60^  per  ounce,  what  is  the  value  of  the  silver  con- 
tained in  a  half-dollar,  which  weighs  192.9  grains,  ^  being 
pure  silver? 

8.  What  per  cent  of  a  lb.  Avoirdupois  is  a  Troy  pound  ? 


286  Chapter  Five. 

MISCELLANEOUS. 

377.  Oral  Problems. 

1.  If  4  books  cost  $  1.25,  what  will  a  dozen  cost  ? 

2.  If  3  pounds  of  sugar  cost  16|-^,  what  will  be  the  cost 
of  50  pounds?  ip0undcosts6^,etc. 

3.  If  48  pounds  of  tea  cost  $  20,  what  will  12  pounds 
COSt  ?  12  pounds  will  cost  \  of  $  20.  * 

4.  Bought  17  yards  of  cloth  for  $  30.  How  many  yards 
could  I  have  bought  for  $  90  ? 

5.  If  36  men  do  a  piece  of  work  in  105  days,  how  long 
will  it  take  72  men  to  do  it  ? 

6.  If  7  railway  trucks  weigh  14  tons,  how  much  would 
29  trucks  weigh  ? 

7.  How  long  will  it  take  8  horses  to  plough  a  field,  if  3 
horses  can  do  it  in  8  days  ? 

8.  What  is  the  height  of  a  steeple  that  casts  a  shadow 
of  300  feet,  if  an  8  foot  pole  casts  a  shadow  of  12  feet. 

9.  If  18  men  mow  90  acres  of  grass  in  5  days,  how  many 
acres  will  36  men  mow  in  5  days  ?     In  10  days  ? 

10.    If  60  yd.  carpet  }  yard  wide  will  cover  a  floor,  how 
many  yards  f  yard  wide  will  be  required  ? 

378.  Written  Problems. 

1.  A  piece  of  cloth,  measured  with  a  yard  measure  that 
is  1  inch  too  short,  appears  to  be  25  yards  long.  What  is 
its  true  length  ? 

2.  Exchanged  40  yd.  muslin,  worth  10J^  per  yard,  for 
15  yards  linen.     What  is  the  value  of  the  linen  per  yard  ? 


Miscellaneous.  287 

3.  If  3  men  or  6  women  can  do  a  piece  of  work  in  56 
days,  in  what  time  will  1  man  and  2  women  working  together 
doit? 

4.  If  5  men  can  do  as  much-  in  a  day  as  8  boys,  how 
long  will  it  take  32  boys  to  finish  a  piece  of  work  which  15 
men  can  do  in  12  days  ? 

5.  If  $  100  gain  $  4  in  1  year,  what  will  $350  gain  in 
Z\  years  ? 

6.  If  48  horses  in  10  days  consume  180  bushels  oats, 
how  many  bushels  will  32  horses  consume  in  10  days  ?  In 
12  days  ?     In  15  days  ? 

7.  If  5  men  mow  45  acres  of  grass  in  6  days,  in  how 
many  days  will  12  men  mow  90  acres  ? 

379.  If  5  men  mow  45  acres  in  6  days, 

1  man  will-  mow  45  acres  in  6  days  x  5. 

1  man  will  mow    1  acre   in  6  days  x  5. 

45 

12  men  will  mow    1  acre  in  6  days  x  5. 

45  x  12 

12  men  will  mow  90  acres  in  6  days  x  5  x  90. 

45  x  12 

Cancelling,  Mays  x  5*  ?0  =  5  days,  Ans. 

I 

380.  In  practice,  the  work  is  somewhat  shortened.  Since  the 
number  of  days  is  required,  we  write  the  given  number  of  days  last, 
with  a  line  underneath. 

5  men  mow  45  acres  .  days. 
1  man  mows  1  acre    I  "  x  5  x  rr. 
12  men  mow  90  acres  J     45  x  12 

If  5  men  do  the  work  in  a  certain  time,  1  man  will  require  5  times 
as  many  days.     We  place  5  in  the  num«rator  (as  a  multiplier).     To 


288  Chapter  Five. 

cut  1  acre,  he  will  take  ^  of  the  time  required  to  cut  45  acres.    Place 
45  in  the  denominator  (as  a  divisor). 

12  men  will  take  T!2  of  the  time  1  man  requires.  Place  12  in  the 
denominator.  To  cut  90  acres  will  require  90  times  as  long.  Place  90 
in  the  numerator. 

8.  If  12  horses  eat  60  bushels  of  oats  in  6  days,  how 
many  bushels  will  24  horses  eat  in  3  days  ? 

Make  bushels  the  last  term. 

12  horses  in  6  days  eat  1  bu# 
1  horse    in  1  day   eats  I  60_ 
24  horses  in  3  days  eat  J 

9.  If  24  men  use  240  pounds  of  beef  in  2  weeks,  how 
many  pounds  will  18  men  use  in  8  weeks  ? 

24  men  in  2  weeks  use  240  lb. 

10.  If  6  printers  can  print  1656  books  in  9  days,  how 
many  books  will  "15  printers  print  in  10  days  ? 

11.  How  much  will  it  cost  to  feed  520  sheep  for  36  days, 
if  it  costs  $  128  to  feed  160  sheep  48  days  ? 

12.  In  what  time  will  8  masons  build  a  wall  84  feet  long, 
working  1.0  hours  a  day,  if  12  masons  build  a  wall  96  feet 
long  in  8  days,  working  8  hours  a  day  ? 

13.  How  much  money  must  I  lend  for  1  year  and 
3  months,  when  the  rate  of  interest  is  5  per  cent,  in  return 
for  $  60  lent  me  for  9  months,  which  I  borrowed  at  4  per 
cent? 

14.  If  27  men  build  54  rods  of  wall  in  6  days,  how  many 
rods  will  32  men  build  in  9  days  ? 

15.  If  50  men  can  do  a  piece  of  work  in  90  days,  working 
8  hours  a  day,  in  how  many  days  will  72  men  do  it,  working 
10  hours  a  day  ? 


Miscellaneous.  289 

16.  If  f  350  earns  $  42  interest  in  3  years,  how  much 
will  $  225  earn  in  5  years  ? 

17.  If  a  wall  34  feet  high  could  be  built  by  68  men  in 
15  days,  how  many  men  could  build  a  wall  32  feet  high  in 
8  days  ? 

18.  If  a  ship's  crew  of  500  men  have  provisions  to  serve 
for  48  days,  at  the  rate  of  27  ounces  a  day  for  each  man, 
how  many  men  will  the  same  provisions  serve  for  60  days, 
allowing  each  man  30  ounces  a  day  ? 

19.  How  many  hours  a  day  must  9  men  work  so  that 
they  may  do  as  much  in  16  days  as  12  men  can  do  in  15  days 
of  8  hours  each  ? 

20.  If  30^  is  paid  for  6  lb.  14  oz.  of  bread,  when  wheat 
is  85^  per  bushel,  what  should  be  paid  for  23  lb.  12  oz., 
when  wheat  is  99^  per  bushel  ? 

21.  If  3  men  can  do  as  much  work  as  7  boys,  how  long 
will  it  take  28  boys  to  do  as  much  work  as  16  men  can  do  in 
24  days  ? 

22.  A  crew  of  16  men  have  provisions  for  36  days,  allow- 
ing 20  ounces  to  each  man  per  day.  After  sailing  10  days 
they  pick  up  10  shipwrecked  sailors.  How  long  will  the 
provisions  then  last  at  the  rate  of  16  ounces  per  man  ? 

23.  If  A  can  do  a  piece  of  work  in  4  days,  and  B  can  do 
the  same  work  in  5  days,  how  many  days  will  it  take  both, 
working  together  ? 

A  can  do  \  of  the  work  in  one  day,  and  B  \  of  it.  Together  they 
can  do  \  +  £,  or  ^  in  one  day.  If  they  do  9  twentieths  in  one  day, 
to  do  20  twentieths,  or  the  whole  work,  will  require  (20  -?-  9)  days,  or 
2|  days. 

24.  If  one  man  can  do  a  piece  of  work  in  24  days,  and 
another  man  can  do  it  in  48  days,  how  long  will  it  take  both, 
working  together  ? 


290  Chapter  Five. 

APPROXIMATIONS. 

Pupils  should  be  drilled  to  take  a  broader  view  of  their  work,  by 
estimating  the  probable  result  before  taking  a  pencil.  In  this  way 
many  absurd  answers  might  be  avoided. 

381.    Give  approximate  answers  at  sight : 

1.  Find  the  interest  of  $  150,  at  4%,  from  Jan.  1,  1903, 
to  Dec.  30,  1905.     (Nearly  3  years.) 

2.  What  is  the  weight,  at  57 J  lb.  per  cubic  feet,  of  a 
cake  of  ice  4  ft.  by  2  ft.  by  1 J  ft  ?  (Nearly  60  lb.  per  cubic 
feet.) 

3.  Find  the  amount  of  goods  sold,  the  commission  at 
2£%  being  $  11.75.     (About  3%.) 

4.  What  %  of  497  is  249  ? 

5.  What  %  of  3ff  is  lift? 

6.  Cost  of  19,987  ft.  boards  at  %  30.05  per  M  ? 

7.  How  much  will  be  paid  for  4  barrels  sugar,  each 
containing  299  pounds,  at  5^  per  pound  ? 

8.  18.0327 -j- 4.5026. 

9.  83«-3ff 

10.  74  A.  155  sq.  rd.  land  at  $  79  per  acre  ? 

11.  487|  is  what  per  cent  of  960  ? 

12.  If  17  bu.  37  lb.  of  corn  cost  $8.75,  what  will  52 
bushels  cost  ? 

13.  About  how  many  cords  of  wood  in  a  pile  25  feet  long, 
4  feet  wide,  5  feet  high  ? 

14.  How  many  bushels  (1\  cu.  ft.)  can  be  placed  in  a  bin 
6  feet  long,  5  feet  wide,  4  feet  high  ? 

15.  How  many  acres  in  a  field  52  rods  long,  30  rods  wide? 

16.  About  how  many  yards  are  there  in  the  side  of  a 
square  field  containing  1  acre  (4840  square  yards)  ? 


Review  of  Simple  Numbers.  291 

REVIEW  OF  SIMPLE  NUMBERS. 

382.  Written  Exercises. 
6748 

X  427  After  multiplying  by  7,  the  pupil  multiplies  this 

47236  latter  product  by  6  tens,  which  gives  him  the  product 

283416  by  42  tens.     In  this  way  one  line  is  saved. 

2881396 

383.  Find  products: 

1.  3,925  x  328  6.  31,265  x  164 

2.  12,345  x  273  7.  5,763  x  426 

3.  2,087  x  287  8.  87,093  x  486 

4.  20,308  x  142  9.  6,905  x  364 

5.  4,321  x  189  10.  64,271  x  357 
3289 

832  First  multiply  by  800,  by  placing  the  first  figure 

26312  of  the  product  by  8  in  the  hundreds'  place.     Multiply 

105248  tnis  Dv  ^>  writing  the  first  figure  in  the  units'  place. 

2736448 

11.  4008  x  214  16.  6352  x  927 

12.  8736  x  742  17.  2781  x  525 

13.  3764  x  327  18.  9060  x  1166 

14.  1087  x  848  19.  6329  x  618 

15.  8319  x  416  20.  2345  x  1272 

21.  Multiply  6984  by  25.  ±  of  698400. 

22.  4327x75 

23.  3762  X  62%.     Multiply  376,200  by  f 

24.  5796  x  62£  27.  7154  x  87£ 

25.  8383  x  12£  28.  6419  x  33J 

26.  3428  x  37J  29.  6208  x  66$ 


292  Chapter  Five. 

REVIEW  OF  FRACTIONS. 


384.   "Written  Exercises. 

7854    xf 

9365     xf 

1963£  Deduct  J. 

1170|  Deduct  f 

5890J  Ans. 

8194f  ^tis. 

Multiply  6578  by  9|. 

65,780    =  10  times  number. 

2,192}  =  i 

number  (deduct). 

63,587J  Ans. 

385.   Find  products: 

1.     176  x  H 

11.   4844  x9£ 

2.     273  x  H 

12.   8960  x  8J 

3.   4554  x  £ 

13.   3245  x  7J 

4.    1001  X  ff 

14.   9060  x  11£ 

5.   3243  x  i 

15.     658  x  99£ 

6.    6776  x  | 

16.     658  x  99f 

7.   2307  x  £ 

17.     725  x  119| 

8.    7284  x  J 

18.     347  x  79f 

9.    5631  x  A 

19.     418  x  89£ 

10.    9657  x  H 

20.     543  x  49J 

386.   Written  Exercises. 

Note.  —  Do  not  use  too  many  figures. 

1.  Add£,2i,f  f. 

2.  Divide  each  of  the  following  fractions  by  6 : 

3.  Keduce  J-  of  -jSj-  of  -fa  of  2-J  to  a  simple  fraction. 

4.  38$-21|f     40$ -18$ 


Review  of  Fractions.  293 

5.  What  fraction  of  £  1  18s.  9d.  is  5s.  6d.  ? 

6.  Multiply  24£  by  f  of  f 

7.  What  is  the  greatest  common  divisor  of  657  and  1168  ? 
the  least  common  multiple  of  12,  16,  20,  30  ? 

8.  What  must  be  taken. from  8^  to  leave  3^  ? 

9.  Eeduce  fff.and  Mf to  their  lowest  terms. 

10.  Which  is  the  greatest  and  which  is  the  least,  -J-  of  ^,  -J 
of  f,  and  2i  of -ft? 

11.  What  must  be  added  to  3ft-  to  make  5f  ? 

12.  Add  f  of  a  week,  f  of  an  hour,  ft-  of  a  minute. 

13.  How  much  is  9  times  each  of  the  following  fractions? 

™„      o    „  -        **  TT'  «*  tr 

14.  3<ty-*-fof7. 

15.  ft  +  fof  ft  +  fof  f. 

16.  What  part  of  a  10-acre  field  is  4  A.  100  sq.  rd.  ? 

17.  What  is  the  least  number  that  will  contain  each  of 
the  numbers  6,  15,  18,  and  20  ? 

18.  What  must  be  multiplied  by  4£  to  produce  16\  ? 

19.  What  is  the  value  of  tilt? 

20.  What  quantity  must  be  divided  by  4^  to  produce  8|  ? 


21.   Find  the  value  of 


gt±j 


22.  How  much  is  i_JI  of  3  da.  15  hr.  32  min.  ? 

23.  Eeduce  ft-  mile  to  rods. 

24.  Add  I,  f,  5£.     Subtract  4ft  from  the  sum. 

25.  Multiply  f  of  5J-  by  7f     Divide  the  result  by  If 


294  Chapter  Five. 

REVIEW  OF  DECIMALS. 

387.   Written  Exercises. 

1.  Express  as  decimals  fifo,  y^,  and  ^. 

2.  .395  +  86.7  +  209.0043  +  .81  +  3.075  +  27. 

3.  Divide  34,020.072  by  5.309.     570  -s-  .005  =  ? 

4.  Multiply  80.037  by  10.     Seventy-three  hundred-thou- 
sandths by  one  hundred.     .2054  x  1000  =  ? 

5.  Subtract  48.8067  from  53.07.     .0539  x  26.08  =  ? 

6.  The  smaller  of  two  numbers  is  8.5307,  and  their  sum 
is  25.07.     Find  the  larger  number. 

7.  Express  .39,  6.175,  .00036,  and  74.0005  as  common 
fractions  (or  mixed  numbers). 

8.  Divide  .826  by  100 ;  543.71  by  10,000 ;  and  fifty-nine 
ten-thousandths  by  one  thousand. 

9.  Find  the  difference  between  9.84  and  38.005,  and  the 
continued  product  of  83.09,  .734,  and  5.007. 

10.  Eeduce  6  shillings  9  pence  to  the  decimal  of  a  pound 
sterling. 

11.  Express  as  decimals  seven  hundredths,  forty-three 
ten-thousandths,  and  ninety-one  millionths. 

12.  Change  y1^,  8^,  y^,  and  jfa  into  decimals.     Find 
their  sum. 

13.  Express  .42796  as  a  common  fraction,  and  the  sum  of 
ft,  dhr>  and.  y$  Jfo-  as  a  decimal. 

14.  3.009  x  .07  x  .0907. 

15.  Divide  .0075  by  .15,  and  .00044408  by  .0112. 

16.  Divisor,  403.6 ;  quotient,  2.709.     Dividend  ? 

17.  What  is  the  value  of  °35  *  QQ56? 

.00007 

18.  Change  69  rods  to  the  decimal  of  a  mile. 

19.  Change  .4285  month  (30  days)  to  days,  hours,  etc. 


Special  Drills.  295 

SPECIAL  DRILLS. 
388.    Give  sums: 

1.  856  +  256  =  856  +  200  +  50  +  6 

The  pupil  says  (or  thinks)  only  1056,  1106,  1112. 

2.  576  +  425  4.   749  +  312  6.    $6.73  +  $3.94 

3.  685  +  599  5.    567  +  658  7.    $8.27  +  $4.89 

Give  remainders : 

8.  1244  -  655  =  1244  -600-50-5 

Think  only  644,  594,  589. 

9.  1021-576       11.   1040-312       13.   $12.00 -$8.73 
10.   1264-685       12.   1322-643       14.   $11.05 -$2.69 

Give  products : 

15.  24  x  21  =  20  times  24  +  24  =  480  +  24 

Say  only  480,  504. 

16.  33  x  21  18.    41  x  41  20.    31  x  31 

17.  22x31  19.   32x41  21.   44x21 

Give  sums : 

22.  425  +  99  =  425  +  100-1 

Say  only  525,  524. 

23.  576  +  99        24.   999  +  425        25.    $8.68 +  $4.99 

Give  remainders : 

26.  565-99  =  565-100  +  1 

Say  only  465,  466. 

27.  743-99      28.    1230-999      29.    $12.13 -$4.99 

Give  products : 

30.  27x99  =  100  times  27 -27  =  2700 -27 

31.  36x99  32.   24x99  33.   98x99 


2<)6  Chapter  Five. 

389.   Oral  Eeview  Problems. 

1.  What  will  be  the  cost  of  48  yards  of  cloth  at  87-^  per 
yard? 

2.  A  horse  was  sold  for  $  80,  which  was  J  of  the  cost. 
How  much  was  lost  on  the  horse  ? 

3.  How  many  yards  of  carpet  27  inches  wide  will  be 
needed  to  cover  a  floor  containing  48  square  yards  ? 

4.  Paid  $3.45  for  groceries,  $1.50  for  dry  goods,  and 
99  ^  for  sundries.     What  is  the  total  ? 

5.  From  a  chest  containing  25  J  pounds  of  tea,  8£  pounds 
were  sold.     How  many  pounds  remain  ? 

6.  What  would  be  the  cost  of  2  bushels  blueberries  at 
5^  per  quart? 

7.  83 J  yards  of  cloth  are  divided  into  9  pieces.  How 
many  yards  are  there  in  each  piece  ? 

8.  I  buy  hardware  to  the  amount  of  $6.37.  I  give  the 
storekeeper  two  $5  bills.  How  much  change  should  I 
receive  ? 

9.  What  will  be  the  cost  of  24  yards  of  calico  at  4|^  per 
yard? 

10.  What  should  I  pay  for  19  baseballs  at  $1.25  each? 

11.  At  $  1.87-J  per  yard,  what  will  be  the  cost  of  120  yards 
of  silk  ? 

12.  For  $120,  how  many  yards  of  silk  can  I  buy  at 
$1.87£  per  yard? 

13.  What  is  the  interest  of  $300,  for  30  days,  at  6  per 
cent? 

14.  What  will  18  oranges  cost  at  35^  per  dozen  ? 

15.  At  4f  ^  per  yard,  how  many  yards  of  calico  can  I  buy 
for  95^? 


Review.  297 

16.  How  many  square  yards  are  there  in  a  field  41  yards 
long,  42  yards  wide  ? 

17.  If  I  pay  15^  for  3J  yards  of  muslin,  what  is  the  price 
per  yard  ? 

18.  How  many  acres  of  land  are  there  in  two  farms  con- 
taining, respectively,  347  and  495  acres  ? 

19.  At  87£^  each,  how  many  baseballs  can  be  bought  for 
$56? 

20.  How  much  will  be  paid  for  21  pounds  butter,  at  28/ 
per  pound  ? 

21.  Paid  23/  for  calico,  27/  for  ribbon,  and  48/  for 
collars.     What  was  the  amount  of  my  bill  ? 

22.  A  farmer  had  95  sheep.     He  sold  39,  and  17  died. 
How  many  had  he  left  ? 

23.  What  will  be  the  cost  of  16  baseballs,  at  49/  each  ? 

24.  How  much  paint  will  there  be  in  27  casks,  each  con- 
taining 75  pounds  ? 

25.  A  man  divided  a  429-acre  farm  into  plots  of  13  acres 
each.     How  many  such  plots  were  there  ? 

26.  There  are  900  men  in  a  certain  regiment.     How  many 
companies  of  75  men  each  are  in  the  regiment  ? 

27.  Find  the  cost  of  136  pounds  sal-soda,  at  \j  per  lb. 

28.  At  19J/  per  yard,  what  will   be  paid  for  64  yards 
gingham  ? 

29.  How  many  square  inches  in  a  sheet  of  paper  10-J- 
inches  long  by  4£  inches  wide  ? 

30.  If  2|  yards  of   cloth  are  needed  for  a  jacket,  how 
many  jackets  can  be  made  from  18|  yards? 

31.  How  many  yards  around  a  field  96  yards  long,  75 
yards  wide  ? 


298  Chapter  Five. 

32.  What  will  be  the  area,  in  square  rods,  of  a  triangle  33 
rods  base,  altitude  42  rods  ? 

33.  How  many  acres  in  4960  square  rods  ? 

34.  How  many  feet  in  a  mile  ? 

35.  I  paid  $16.25  for  cloth  at  $1.25  per  yard.  How 
many  yards  did  I  buy  ? 

36.  Half  a  number  -f  ^  of  the  same  number  =  85.  What 
is  the  number  ? 

37.  I  mix  4  pounds  of  coffee  costing  20^,  with  6  pounds 
costing  25^.     What  is  the  mixture  worth  per  pound  ? 

38.  A  tailor  makes  up  99  yards  of  cloth  into  trousers, 
using  2|  yards  per  pair.  How  many  pairs  of  trousers  does 
he  make  ? 

39.  At  60^  per  pound,  what  will  be  the  cost  of  a  chest  of 
tea  weighing  45  pounds  ? 

40.  A  man  owns  a  strip  of  land  with  a  frontage  of  576 
feet.     How  many  lots  18  feet  front  can  he  make  ? 

41.  A  can  do  a  piece  of  work  in  5  hours,  B  in  7  hours. 
How  long  will  it  take  both  working  together  ? 

42.  At  what  rate  will  $  300  gain  $  24  in  2  years  ? 

43.  What  sum  of  money  will  gain  $30,  in  2  yr.  6  mo.,  at 
6%? 

44.  If  a  staff  i2  feet  long  casts  a  shadow  of  3  feet,  what 
is  the  length  of  a  pole  that  casts  a  shadow  of  27  feet  at  the 
same  time  ? 

45.  If  20  men  can  perform  a  piece  of  work  in  8  days, 
how  many  men  will  it  take  to  do  the  same  work  in  5  days? 

46.  An  agent  receives  $8200  to  invest  after  deducting 
his  commission  of  -fa  of  the  amount  invested.  What  is  the 
agent's  commission? 

47.  A  lot  is  sold  for  $1200,  at  a  loss  of  20  per  cent 
What  part  of  $  1200  is  the  loss  ? 


Review.  299 

390.   Written  Problems. 

1.  A  rug  costs  $  20.  It  is  sold  at  a  profit  of  20%.  The 
selling  price  is  20%  below  the  marked  price.  How  much  is 
received  for  the  rug  ?     What  is  the  marked  price  ? 

2.  What  price  must  cloth,  which  cost  $  2  per  yard,  be 
marked  so  that  a  profit  of  20  %  will  be  made  when  the  cloth 
is  sold  at  20  %  less  than  the  marked  price  ? 

3.  A  coal  bin  is  6  feet  long  and  4  feet  wide.  How  deep 
must  it  be  to  contain  5  tons  of  stove  coal,  if  one  ton  occupies 
36  cubic  feet  of  space  ? 

4.  A  man  walking  at  the  rate  of  3  mi.  96  rd.  per  hour 
will  walk  how  far  in  3  hr.  16  min.  ? 

5.  If  a  merchant  pays  6\tf  per  yard  for  muslin,  and  sells 
the  same  for  7\  $  per  yard,  what  is  his  gain  per  cent  ? 

6.  Make  and  solve  a  problem  illustrating  the  application 
of  percentage  to  the  finding  of  an  agent's  commission. 

7.  Multiply  eight  hundred  (units)  and  forty-six  ten- 
thousandths  by  three  thousand  forty  millionths. 

8.  What  is  the  interest  on  $  128.40,  for  1  yr.  5  mo.  17  da. 
at  6  per  cent  ? 

9.  A  regiment  of  940  men,  during  the  war,  lost  532  of 
their  number  by  death  and  125  by  desertion.  What  was 
the  percentage  of  loss  in  each  case,  and  what  per  cent 
remained  for  service  ? 

10.  A  merchant  sold  a  lot  of  damaged  sugar  at  a  loss  of 
25  per  cent,  receiving  $  1972.65.  How  much  did  the  sugar 
cost  him  ? 

11.  What  is  a  pile  of  wood  15  feet  long,  10£  feet  high, 
and  12  feet  wide  worth,  at  $  4^-  per  cord  ? 

(1  cord  =  128  cu.  ft.) 

12.  Add  the  greatest  and  the  least  of  the  three  fractions 
H>  h  if  5  an^  divide  the  sum  by  the  remaining  fraction. 


joo  Chapter  Five. 

13.  Multiply  82  ten-thousandths  by  7  and  5  hundredths, 
and  divide  the  product  by  705  millionths. 

14.  Find  the  cost  of  96  feet  of  pine  lumber  at  $  25  per  M, 
and  1650  laths  at  $  3  per  M. 

15.  A  horse  costing  $  160  is  sold  for  $  180.  What  is  the 
gain  per  cent  ?  What  is  the  loss  per  cent  when  a  horse 
costing  $  180  is  sold  for  $  160  ? 

16.  A  merchant  sold  600  barrels  of  flour  for  $  3450,  at 
a  loss  of  4J  per  cent.  What  did  the  flour  cost  him  per 
barrel  ? 

17.  How  long  would  it  take  a  person  to  count  a  million 
silver  dollars,  at  the  rate  of  100  a  minute,  and  working 
8  hours  a  day? 

18.  Find  the  number  of  days  from  March  2,  1903,  to 
August  11,  1903. 

19.  Find  the  interest  on  a  note  for  $  250,  dated  Jan.  21, 
1904,  and  paid  May  30,  1904,  at  6  %. 

20.  Divide  22.5  by  51.75,  and  express  the  result  in  the 
form  of  a  fraction. 

21.  By  the  census  of  1890,  the  population  of  a  certain 
city  was  26,275.  By  the  census  of  1900,  its  population  was 
31,530.     Find  the  per  cent  of  increase. 

22.  Each  of  two  boys  bought  100  apples  for  a  dollar. 
The  first  boy  sold  his,  4  apples  for  5^ ;  the  second  sold  his, 
5  apples  for  6$.  Which  boy  gains  the  more  per  cent? 
How  much  more  ? 

23.  A  quantity  of  coal  was  bought  for  $900.  For  what 
must  it  be  sold  to  gain  33J  %  ? 

24.  By  selling  a  house  for  $5760,  a  man  gained  on  the 
cost  25  %.     What  was  the  cost  ? 

25.  Change  to  other  methods  of  expression,  J,  -J-,  .37^,  J, 
.16J. 


Review.  301 

26.  A  note  of  $  1260,  dated  July  5, 1904,  was  paid  June  7, 
1906,  with  interest  at  8%.     What  was  the  amount  paid  ? 

27.  A  flock  of  sheep  has  been  increased  by  250%  of  its 
number,  and  now  numbers  1050.  What  was  the  original 
number  ? 

28.  Bought  a  house  for  $  6240,  and  sold  it  so  as  to  gain 
35%.     What  did  I  sell  it  for  ? 

29.  Sold  goods  at  a  loss  of  20%,  an  actual  loss  of  $  57.50. 
What  was  the  first  cost  ? 

30.  The  milk  from  a  herd  of  25  Jersey  cows,  sold  at  6  $ 
a  quart,  amounted  in  one  summer  to  $2025.  How  many 
quarts  were  sold,  and  what  was  the  average  quantity  from 
each  cow  ? 

31.  A  woman  has  three  children.  She  pays  for  each  $  15 
a  year  for  having  his  clothes  made,  $  1.50  a  month  for  his 
mending,  and  $  0.35  a  week  for  his  washing.  How  much 
could  she  save  in  a  year  if  she  knew  how  to  wash,  make 
clothes,  and  mend  ? 

32.  A  farmer  exchanged  340  bushels  of  corn  worth  75^ 
per  bushel,  for  barley  worth  $  1  per  bushel,  and  oats  worth 
50  ^  per  bushel.  How  many  bushels  of  each  did  he  receive, 
the  quantity  of  barley  and  oats  being  equal  ? 

33.  A  pole  stands  \  in  the  mud,  f  in  the  water,  and  32  ft. 
in  the  air.     How  long  is  the  pole  ? 

34.  Bought  flour  for  $  8.25,  and  sold  it  for  $  9.  What  is 
the  per  cent  of  gain  ? 

35.  Bought  flour  for  $  9  and  sold  it  for  $  8.25.  What  is 
the  per  cent  of  loss  ? 

36.  If  two-thirds  of  a  yard  of  silk  can  be  bought  for  $f, 
how  many  yards  can  be  bought  for  $  3|  ? 

37.  A  drover  sold  250  sheep  for  $1150,  which  was  15% 
more  than  they  cost.     What  was  the  cost  of  each  sheep  ? 


302  Chapter  Five. 

38.  Find  a  common  divisor  of  72  and  90. 

39.  How  many  feet  of  paper,  18  inches  wide,  will  paper 
the  sides  of  a  room  16  feet  by  14  feet,  and  10  feet  high,  de- 
ducting 174  square  feet  for  doors  and  windows  ? 

40.  Find  the  sum  of  fa  $,  f§,  -^,  ^-,  in  decimals,  correct 
to  fourth  place. 

41.  The  dividend  is  9876,  the  quotient  is  87,  the  remain- 
der is  45.     Find  the  divisor. 

42.  Change  .03125  to  a  common  fraction  in  smallest 
terms. 

43.  Bought  a  hogshead  of  sugar  containing  848  pounds  for 
$  38.16,  and  paid  $  4.24  freight  and  cartage.  At  what  price 
per  pound  must  it  be  sold  to  gain  20  %  ? 

44.  To  f  of  f  add  \  of  fa,  and  reduce  to  lowest  terms ; 
multiply  the  sum  so  obtained  by  If,  and  reduce  to  a  mixed 
number ;  from  the  product  subtract  f ,  and  reduce  to  lowest 
terms ;  divide  the  remainder  by  5,  and  convert  the  quotient 
into  a  decimal  fraction;  add  1.1 ;  multiply  by  2.5;  subtract 
.9 ;  and  divide  the  remainder  by  .007. 

45.  A  can  weigh  a  certain  quantity  of  goods  in  15  days 
by  working  7  hours  a  day.  How  long  will  it  take  him  to 
do  the  same  work  by  working  9  hours  a  day  ? 

46.  In  an  example  in  division  the  remainder  is  14,  the 
divisor  is  16,  and  the  quotient  is  18.     What  is  the  dividend  ? 

47.  Solve  by  cancellation : 

How  many  pieces  of  cotton  cloth,  each  piece  containing 
42  yards,  at  9-J-  ^  per  yard,  can  be  bought  for  14  firkins  of 
butter,  each  containing  56  pounds,  at  19^  per  pound  ? 

48.  What  must  be  the  depth  of  a  bin  which  is  4  ft.  wide 
and  6  ft.  long,  to  contain  40  bushels  oats  ? 

49.  A  farmer  sold  9875  pounds  hay  at  $  12£  per  ton,  and 
took  in  part  payment  5000  feet  of  boards  at  $11  per 
thousand.     How  much  remained  due  him  ? 


Review. 


303 


50.  Bought  80  barrels  of  flour  at  $  6  per  barrel,  paying 
for  freight  $  30.  At  what  price  must  I  sell  it  per  barrel  to 
gain  30  %  on  the  total  cost  ? 

51.  What  is  the  amount  of  $  720.50,  for  3  yr.  5  mo.  19 
da.,  at  6  per  cent  ? 

52.  Three  men  buy  a  house  for  $  2500.  A  pays  $  500, 
B  pays  $  900,  C  pays  f  1100.  They  rent  it  for  $  250. 
What  is  each  one's  share  of  the  rent  ? 

53.  If  12.875  acres  of  land  cost  $  1030,  what  will  4.75 
acres  cost  ? 

54.  Write  three-fourths  of  one  per  cent,  first  as  a  pure 
decimal,  and  again  as  a  common  fraction. 

55.  If  a  man  paid  $18f  for  a  load  of  hay  weighing  1£ 
tons,  what  would  he  pay  at  the  same  rate  for  f  of  a  ton  ? 

56.  If  11  weavers  in  9  days  weave  1584  yards,  what  will 
1  man  do  in  1  day  ?    6  men  in  7  days  ? 

57.  What  is  the  exact  interest  of  $  500,  for  100  days,  at 
8  per  cent  ?     (Take  365  days  to  the  year.) 

58.  Divide  the  product  of  8|  and  llf  by  their  difference. 

59.  A  merchant  bought  340  bushels  of  potatoes  at  80^ 
per  bushel ;  20  per  cent  of  them  proved  worthless,  and  were 
thrown  away.  He  sold  the  remainder  at  $  1.10  a  bushel. 
What  did  he  gain  or  lose  ? 

60.  Divide  eighty-four  and  eighty-four  hundredths  by 
forty-eight  thousandths. 

61.  How  much  money  in  silver  dollars,  41 2  J-  grains  each, 
will  weigh  165  pounds  Avoirdupois,  7000  grains  to  the 
pound  ? 

62.  What  is  the  amount  of  f  1395,  at  4  per  cent,  for 
7  mo.  24  da. 

63.  A  coal  dealer  buys  150  tons  of  coal,  2240  pounds 
each,  at  $  4.50  per  ton.  He  sells  it  at  f  4.75  per  ton,  giving 
2000  pounds  to  the  ton.     What  is  his  profit  ? 


304  Chapter  Five. 

64.  What  is  the  value  of  (J  of  f  of  3f +8£)-^(10£-7f£)  ? 

65.  How  many  bushels  of  grain  will  fill  a  bin  8.5  feet 
long,  4.25  feet  wide,  and  3|  feet  deep  ? 

66.  Three  workmen  receive  $  283.50  for  doing  a  piece  of 
work.  One  worked  32  days,  the  second  worked  53  days,  the 
third  worked  41  days.     What  is  the  share  of  each  ? 

67.  A  man  bought  silverware  for  $  120,  and  sold  it  for 
$  250  less  33£  and  10  per  cent.  What  was  his  profit  per 
cent? 

68.  What  is  the  interest  on  f  356.75,  at  4  per  cent,  for 
3  yr.  5  mo.  14  da.  ? 

69.  A  note  for  $  600,  drawn  Jan.  16,  payable  4  months 
after  date,  is  discounted  March  25  at  a  bank,  at  6  per  cent. 
What  are  the  proceeds  ? 

70.  A  dry-goods  merchant  sells  goods  12J^  per  yard  more 
than  their  cost,  and  realizes  a  profit  of  8  per  cent.  What  is 
the  cost  per  yard  ? 

71.  A  man  bought  396  acres  of  land  for  $40,293.  He 
sold  150  acres  at  $  120  per  acre,  134  acres  at  $  80  per  acre, 
and  the  remainder  at  cost.  Did  he  gain  or  lose,  and  how 
much  ? 

72.  If  44f  yards  of  calico  cost  $  1.99,  how  much  must  be 
paid  for  80  yards  ? 

73.  Divide  the  sum  of  75  thousandths  and  75  ten-thou- 
sandths by  the  difference  between  75  hundredths  and  75 
tenths. 

74.  What  number  divided  by  320  gives  47  for  quotient 
and  163  for  remainder? 

75.  In  a  schoolroom  there  are  35  pupils  and  a  teacher. 
The  room  is  30  feet  long,  20  feet  wide,  and  15  feet  high. 
How  many  cubic  feet  of  air  space  has  each  person  ? 

76.  A  merchant  sold  a  quantity  of  flour  for  $  282,  losing 
6  per  cent.     How  much  money  did  he  lose  ? 


Review.  305 

77.  I  bought  2500  bushels  of  wheat  at  80^  per  bushel, 
and  sold  it  for  84  f  per  bushel,  on  a  note  for  60  days,  which 
I  had  discounted  immediately  at  a  bank,  at  6  %.  How  much 
did  I  gain  ? 

78.  A  merchant  bought  84  yards  of  linen  at  55^  per  yard, 
and  105  yards  of  muslin  at  20^  per  yard.  He  sold  all  the 
linen  at  40^  per  yard.  What  must  he  charge  per  yard  for 
the  muslin  in  order  to  make  up  exactly  his  loss  on  the  linen  ? 

79.  A  fruit  dealer  bought  a  lot  of  oranges  for  $  240.  He 
sold  \  of  them  for  \  of  the  entire  cost ;  \  of  the  remainder 
for  I  of  the  entire  cost ;  \  of  what  then  remained  for  \  of 
the  entire  cost ;  and  the  final  remainder  for  \  of  the  entire 
cost.     What  was  his  gain  or  loss? 

80.  The  owner  of  165  shares  of  gas  stock  sold  them  at 
$  25  per  share,  and  with  the  proceeds  purchased  two  lots, 
32  feet  by  115  feet,  and  30  feet  by  105  feet,  respectively, 
and  had  just  $  27  left.  What  was  the  price  per  square  foot 
of  the  lots? 

81.  *A  man  purchased  a  house,  paying  for  it  in  four  pay- 
ments as  follows :  on  the  first  payment  \  of  the  purchase 
price;  on  the  second  payment  \  of  the  remainder;  on  the 
third  payment  f  of  what  then  remained  due;  and  on  the 
last  payment  $  2000.  What  was  the  full  amount  paid  for 
the  house  ? 

82.  Find  the  difference  between  the  greatest  common 
divisor  of  480  and  520,  and  the  least  common  multiple  of 
5,  6,  15,  and  20. 

83.  Find  the  value  of  a  pile  of  wood  40  feet  long,  8  feet 
wide,  and  4  ft.  6  in.  high,  at  $5.50  a  cord. 

84.  A  cargo  of  flour  was  bought  for  $690.  For  what 
must  it  be  sold  to  gain  66f  %  ? 

85.  Find  the  sum  of  all  the  prime  numbers  to  50. 


306  Chapter  Five. 

86.  If  A  and  B  can  mow  a  field  in  seven  days,  and  A,  B, 

and  C  mow  it  in  five  days,  for  $  25,  what  ought  C  to  receive  ? 

87.  To  f  of  a  score  add  f  of  a  dozen,  and  from  the  sum 
subtract  f  of  a  hundred.     What  is  the  remainder  ? 

88.  What  must  be  the  length  of  a  load  of  wood  that  is 
4  feet  wide  and  5  J  feet  high  to  contain  2  cords  ? 

89.  Bought  a  hogshead  of  molasses  containing  128  gallons, 
at  65  $  a  gallon ;  paid  80  ^  for  cartage,  and  lost  16  gallons 
by  leakage.  At  what  price  per  gallon  must  the  remainder 
be  sold  to  gain  one-fifth  of  the  entire  cost  ? 

90.  What  is  the  least  number  that  will  exactly  contain 
48,  20,  21,  24  ? 

91.  Sold  50  sofas  for  $2250.  25  of  them  were  sold  at  a 
gain  of  20  per  cent,  and  25  at  a  loss  of  20  per  cent.  What 
was  the  gain  or  loss  on  the  transaction  ? 

92.  Bought  a  number  of  eggs,  and  sold  11  of  them  for 
what  18  cost  me.     What  was  my  gain  per  cent  ? 

93.  A  bookseller  wishes  to  mark  up  the  price  of  a  book 
which  he  is  now  selling  for  $2,  so  that  he  can  deduct  15  per 
cent,  and  yet  receive  the  present  price.  What  must  be  the 
marked  price  ? 

94.  What  is  the  difference  between  .75  divided  by  75, 
and  75  divided  by  .75  ? 

95.  A  watch  that  loses  35  seconds  in  an  hour  was  set 
right  at  noon  on  Monday.  What  time  did  it  show  at  6  p.m. 
the  following  Thursday  ? 

96.  Mr.  A.  sold  a  horse  for  $  240,  which  was  20  per  cent 
less  than  he  asked  for  it,  and  his  asking  price  was  20  per 
cent  more  than  the  horse  cost  him.  What  was  the  cost  of 
the  horse  ? 

97.  Three  quarts  dry  measure  is  what  per  cent  of  a 
bushel  ? 


Review.  307 

98.  What  will  it  cost  to  carpet  an  office  room  measuring 
21  feet  in  length,  and  18  feet  in  width,  the  carpeting  being 
§  yard  wide,  and  costing  $  1.35  per  lineal  yard  ? 

99.  A  physician  accepts,  in  payment  of  a  bill,  a  note  for 
$275.75,  due  in  one  year  and  three  months,  with  interest 
at  7  per  cent.     What  amount  will  be  due  at  maturity  ? 

100.  At  what  rate  will  $1500  amount  to  $1684.50,  in 
2  yr.  18  da.  ? 

101.  How  shall  I  mark  goods  that  cost  me  $.96  a  yard, 
in  order  to  abate  15%  and  still  make  15%  ? 

102.  What  will  it  cost  to  insure  a  factory  valued  at 
$21,000,  at£%,  and  the  machinery  valued  at  $15,400, 
at  f  %  ? 

103.  In  what  time  will  $750  gain  $195  interest,  at  4%  ? 

104.  What  is  the  rate  per  cent  when  the  amount  of  $500 
is  $593.75,  for  2  yr.  and  6  mo.  ? 

105.  What  principal  will  gain  $360  in  5  yr.  4  mo.,  at 

4£%? 

106.  Bought  480  barrels  of  flour,  at  $4.50  a  barrel,  and 
sold  it  for  $2880.     Find  the  gain  per  cent. 

107.  By  selling  a  house  for  $10,304,  a  man  gained  15% 
on  the  cost.     What  was  the  cost  ? 

108.  A  man,  dying,  left  -J  of  his  estate  to  his  wife,  §  of 
the  remainder  to  his  son,  and 'the  remainder  to  his  daughter, 
who  received  $5000.  What  was  the  value  of  the  estate, 
and  what  was  the  son's  share  ? 

109.  What  is  the  interest  of  $10,  for  10  yr.  10  mo.  10  da., 
at  10  per  cent  ? 

110.  If  it  takes  one  man  1\  days  to  do  a  piece  of  work, 
how  long  will  it  take  3  men  to  do  2J  times  as  much  ? 


308  Chapter  Five. 

111.  A  grocer  pays  18^  per  pound  for  coffee,  and  roasts 
it,  losing  10%  of  the  weight  in  the  process.  What  must  he 
charge  per  pound  for  the  roasted  coffee  in  order  to  make  a 
profit  of  20%  ? 

112.  A  merchant  bought  48  bales  of  cotton,  and  then  sold 
the  lot  for  $2008.80,  losing  7%.  What  was  the  cost  per 
bale  ? 

113.  What  is  the  cost  of  sawing  a  pile  of  wood  20  feet  long, 
4  feet  wide,  and  6  feet  high,  at  $  1.20  a  cord  ? 

114.  After  increasing  the  wages  of  his  workmen  33^%,  a 
manufacturer  paid  them  $2.60  a  day.  What  did  he  pay 
them  before  ? 

115.  What  should  a  bookseller  charge  for  a  book  for 
which  he  paid  at  the  rate  of  $  54  a  dozen,  that  he  may  make 
20%  on  the  cost? 

116.  What  is  the  per  cent  profit  or  loss  when  a  hundred 
logs  which  cost  $  65  are  sold  at  78  ^  each  ? 

117.  A  man  spent  ^-,  and  invested  in  his  business  ^,  of 
his  income.  He  deposited  the  remainder,  $  1850,  in  a  bank. 
What  was  his  income  ? 

118.  Sold  a  horse  for  $  322,  and  thereby  lost  8%.  WThat 
should  I  have  sold  it  for  to  gain  15%  ? 

119.  Bought  a  horse  for  $340;  paid  $60  for  keeping 
him,  and  then  sold  him  for  $  540.  What  per  cent  was 
gained  ? 

120.  John  bought  Yl\  pounds  of  sugar  at  h\f  a  pound, 
spending  25%  of  his  money.     How  much  had  he  at  first  ? 

121.  When  10.25  bushels  of  wheat  cost  $  12.71,  what  will 
7J  bushels  cost  ? 

122.  Mr.  Jones  paid  $  15.12  for  the  use  of  a  sum  of  money 
for  1  yr.  6  mo.,  at  5%.     What  was  the  sum  ? 

123.  What  were  the  proceeds  of  a  note  for  $  725.14,  due 
July  7,  discounted  at  a  bank  June  20,  at  8%  ? 


Review.  309 

124.  After  Mr.  Jones  had  spent  37|%  of  his  money,  he 
found  that  he  then  had  enough  to  buy  80  pounds  of  rice  at 
6i$  a  pound.  How  much  could  he  have  bought  with  the 
whole  of  his  money  ? 

125.  On  the  10th  day  of  November,  1899,  you  lent 
William  Rogers  $  864.50.  How  much  does  he  owe  you  to- 
day, the  rate  of  interest  being  4-J  %  ? 

126.  A  man  bought  wheat  for  $10,867,  and  sold  it  at  a 
gain  of  4|%.     What  did  he  receive  for  it  ? 

127.  Divide  three  million  by  six  thousand,  and  multiply 
the  quotient  by  .024. 

128.  How  much  must  I  have  invested  at  5  %  that  my  in- 
come may  be  $  2880  per  year  ? 

129.  Add  these  across,  placing  the  totals  in  the  space  in- 
dicated ;  then  add  the  totals  : 


Totals. 

14,305 

10,702 

18,346 

37,946 

43,865 

17,387 

22,324 

17,437 

18,438 

3,741 

22,972 

25,960 

13,849 

67,431 

34,965 

12,674 

32,905 

1,468 

15,607 

27,865 

32,476 

18,430 

33,301 

18,695 

19,898 

13,460 

27,686 

23,492 

13,852 

26,973 

130.  If  1998,  or  27  per  cent,  of  the  inhabitants  of  a  town 
are  voters,  how  many  inhabitants  has  the  town  ? 

131.  Ten  cows  were  sold  for  $  690,  at  a  gain  of  15  per 
cent.  For  how  much  per  head  on  the  average  should  they 
have  been  sold  to  gain  20  per  cent  ? 

132.  Find  the  interest  of  $  575.50,  for  1  yr.  10  mo.  15  da., 
at  5%. 


CHAPTER  VI. 

PAGES 

Ratio  and  Proportion 310  to  328 

Ratio,  Proportion,  Partitive  Proportion,  Partnership, 
Compound  Proportion. 

Involution  and  Evolution 328  to  338 

Square  Root,  Applications  of  Square  Root,  Cube  Root. 

Mensuration 339  to  357 

The  Circle,  Areas  of  Circles,  Areas  of  Triangles,  Areas 
of  Quadrilaterals,  Surfaces  of  Prisms  and  Cylinders, 
Surfaces  of  Pyramids  and  Cones,  Volumes  of  Prisms 
and  Pyramids,  Volumes  of  Cylinders  and  Cones,  Sur- 
face of  Sphere,  Volume  of  Sphere,  Circular  Measure. 

Longitude  and  Solar  Time 358  to  363 

Standard  Time,  Solar  Time. 

Review  Problems 363  to  366 

Miscellaneous,  Oral,  Written. 

Stocks  and  Bonds 367  to  372 

Domestic  Exchange 373  to  377 

Sight  Drafts,  Time  Drafts,  Bills  of  Exchange. 

Interest 378  to  380 

Compound  Interest,  Annual  Interest. 

Metric  System 380  to  384 

Review  Problems 384  to  414 

Special  Drills,  Review  of  Fractions,  Review  of  Denomi- 
nate Numbers,  Review  of  Commercial  Discount,  Review 
of  Interest,  Review  of  Bank  Discount,  Exact  Interest, 
Miscellaneous  —  Oral  and  Written. 

RATIO. 

391.   Ratio  is  the  relation  which  one  number  has  to  another 
of  the  same  kind. 

The  sign  of  ratio  is  the  colon  (:). 

The  ratio  of  3  to  6  is  expressed  3 :  6. 

The  colon  (:)  is  the  sign  used  in  France  and  Germany  to  indicate 

division  as  well  as  ratio. 

310 


Ratio.  311 

).  The  terms  of  the  ratio  are  the  numbers  compared,  the 
first  being  called  the  antecedent,  and  the  second  the  conse- 
quent.    Both  terms  constitute  a  couplet. 

The  ratio  of  3  to  6  is  obtained  by  dividing  the  antece- 
dent by  the  consequent ;  3 :  6  means  f ,  which  is  equal  to  £. 

393.  Oral  Exercises. 
Find  the  ratio  of : 

1.  175  to  700.  m  =  h  Ans. 

2.  $  36.50  to  $  18.25.        §|^?  =  2.  Ans. 

%  18.25 

Note.  —  The  quotient  is  abstract. 

3.  6  pecks  to  5  bushels.        6  peclf  =  A-  Ans. 

Note.  — The  antecedent  and  the  consequent  must  be  like  numbers, 

4.  1 19  to  $  95.  6.    7  tenths  to  3  fifths. 

5.  20  mills  to  1  dollar.  7.   3  quarts  to  4  gallons. 
8.   1  gallon  to  500  cubic  inches. 

394.  Written  Problems. 

1.    One  line  is  3  rd.  4  yd.  long;  the  length  of  another  is 
5  rd.  1  ft.     Find  the  ratio  of  the  first  to  the  second. 

The  antecedent  3  rd.  4  yd.  is  to  be  divided  by  the  consequent 

5  yd.  1  ft.     As  the  divisor  and  the  dividend  must  be  like  numbers, 

both  terms  of  the  couplet 

are  reduced  to  feet.     The     3  rd.  4  yd.  =  61^  ft.  _  123    j^ 

division  is  indicated  by  writ-      *  ro-*  1  "•       °3 J  ft.      167 

ing  the  antecedent  above  the 

consequent  as  a  fraction.     The  concrete  fraction  — 2 — 1  \s  changed  to 

83*  ft. 

the  abstract  complex  fraction  — ■ ,  which  is  reduced  to  a  simple  f rac- 

88| 

tion  by  multiplying  both  terms  by  2,  giving  |  £^  for  the  result. 

Make  the  antecedent  and  the  consequent  like  numbers,  and 
divide  the  former  by  the  latter. 


312  Chapter  Six. 

2.  M  walks  in  1  hr.  47  min.  as  far  as  N  walks  in  2  hr. 
■3  min.     What  is  the  ratio  of  M's  speed  to  N's  ? 

In  this  example  is  required  the  ratio  of  M's  speed  to  N's.  The 
antecedent  is,  therefore,  M's  speed,  and  the  consequent  is  N's  speed. 
As  the  distance  walked  is  not  given,  x  may  be  used  to  represent  the 
number  of  feet  or  yards  or  miles  walked  by  M  in  107  minutes,  and  by 
N  in  123  minutes.     -^-  will  represent  the  distance  walked  by  M  in 

1  minute,  or  M's  speed,  and  -^- ,  N's  speed.    The  ratio  of  M's  speed 

to  N's  will  be  JL  +  JL    or  —  x— •    Cancelling  x  in  each,  the 
107      123'        107       3  & 

result  is  {ffi ,  or  1^.  Ans. 

3.  One  candle  lasts  4  hr.  20  min.;  another  lasts  3  hr. 
15  min.     Find  the  ratio  of  the  first  to  the  second. 

4.  A  pound  of  coffee  costs  25 J  f ;  1  pound  of  sugar  costs 
&A  ^     What  is  the  ratio  of  price  of  sugar  to  that  of  coffee  ? 

5.  P  earns  in  19f  days  as  much  as  Q  in  18|  days.  What 
is  the  ratio  of  Q's  daily  earnings  to  P's  ?     Of  P's  to  Q's  ? 

6.  One  wheel  makes  600  revolutions  in  8^  seconds;  a 
second  makes  300  revolutions  in  3\  seconds.  What  is  the 
ratio  of  the  speed  of  the  first  wheel  to  that  of  the  second  ? 

7.  The  circumference  of  a  circle  is  12.5664  feet,  and  its 
radius  is  2  feet.  What  is  the  ratio  of  the  diameter  to  the 
circumference  ? 

8.  One  train  goes  40  miles  in  50  minutes ;  another  goes 
24  miles  in  a  half  hour.  What  is  the  ratio  of  the  speed  of 
the  second  to  that  of  the  first  ? 

Find  the  number  of  miles  each  goes  in  an  hour. 

9.  One  window  is  6  ft.  8  in.  by  4  ft.  2  in.;  a  second  is 
4  ft.  8  in.  by  2  ft.  1  in.  What  is  the  ratio  of  the  area  of  the 
second  to  that  of  the  first  ? 

(4f  X  2^)  +  (6f  x  H) 


Ratio.  3I3 

10.  A  mother  is  now  35  years  old,  and  her  son  is  3  years 
and  6  months  old.  Fourteen  months  ago  what  was  the 
ratio  of  the  mother's  age  to  that  of  her  son  ? 

11.  A  farm  costing  $  4750  was  sold  for  $  5750.  What  is 
the  ratio  between  the  profit  and  the  cost  ? 

12.  A  man  can  do  a  piece  of  work  in  4^  days.  What 
part  of  it  can  he  do  in  a  day  and  a  half  ?  What  decimal  ? 
What  per  cent  ? 

13.  What  is  the  ratio  between  a  ton  of  2000  pounds  and 
one  of  2240  pounds  ? 

395.   Oral  Problems. 

1.  One  line  is  a  rod  long;  another  is  5|-  ft.  long.  What 
is  the  ratio  of  the  first  to  the  second  ? 

2.  What  is  the  ratio  of  7  hours  to  one  day  ? 

3.  A  pound  of  coffee  costs  30^,  of  sugar  6^.  What  is 
the  ratio  of  their  respective  prices  ? 

4.  A  walks  in  4  hours  as  far  as  B  in  5.  What  is  the 
ratio  of  A's  speed  to  B's  ? 

5.  E  earns  in  6  days  as  much  as  D  earns  in  8  days. 
Find  the  ratio  of  E's  daily  earnings  to  D's. 

6.  One  wheel  makes  300  revolutions  in  2  minutes ;  the 
second  requires  only  1J  minutes  to  make  the  same  number. 
Find  the  ratio  of  the  number  of  revolutions  made  by  the 
first  wheel  in  1  minute  to  the  number  made  by  the  second 
wheel  in  the  same  time. 

7.  A  circle  whose  diameter  is  1  foot  has  a  circumference 
of  3|  feet.  What  is  the  ratio  of  the  diameter  to  the  circum- 
ference ? 

8.  One  train  goes  40  miles  an  hour ;  a  second  goes  45 
miles  an  hour.  What  is  the  ratio  of  the  speed  of  the  first 
to  that  of  the  second  ? 


314  Chapter  Six. 


17 
21" 

_51 

18  _ 

36 

? 

"70* 

? 
24  = 

57 

'72* 

$16   7 

marks 

? 

21  marks 

5  + 

22  = 

?-f-88. 

PROPORTION. 

396.  Preliminary  Exercises. 

1.  A  =  l. 

16      64 
2     18-36 

3.    »..£ 

13     65 

•        lj^l? 
'   3bu.      $24* 

3qt1  =  30^> 
lgal.      ?** 

11.  6  horses  -j-  ?  horses  =  $  600  -r-  $  900. 

12.  lft. -5-?  yd.  =  15^-*- 90*. 

13.  1  qt.  lpt.  -*-l  pt.  =  ?jt  +  4? 

14.  li+i^JH-f. 

15.  2.8 -r- .4  =  .14 -h  #. 

397.  Two  equal  ratios  form  a  proportion. 

The  ratio  of  3  to  9  is  \,  which  is  also  the  ratio  of  13  to  39. 
This  may  be  expressed  -|  =  Jf ,  or  3  :  9  =  13  :  39.  Substitut- 
ing a  double  colon  (:  :)  for  the  sign  of  equality  (=),  we 
have  the  following  proportion : 

3  :  9  :  :  13  :  39. 

This  is  read,  3  is  to  9  as  13  is  to  39. 

In  the  foregoing  proportion,  3  and  13  are  the  antecedents,  and  9 
and  39  are  the  consequents. 

398.  The  first  and  the  last  term  of  a  proportion  constitute 
the  extremes  ;  the  second  and  the  third  the  means. 

In  the  following  proportion 

5:15:: 9:27 
6  and  27  are  the  extremes,  15  and  9  are  the  means. 


Proportion.  315 

The  foregoing  proportion  may  be  written 

ft-* 

Multiplying  each  of  these  two  fractions  by  the  product 
of  the  denominators,  15  x  27,  we  have 

5x  ?5x27^9  x!5xgT 

u  » 

Cancelling,  5  x  27  =  9  x  15. 

In  the  same  way  it  may  be  shown  that  in  any  proportion 
the  product  of  the  numbers  in  the  extremes  is  equal  to  the 
product  of  the  numbers  in  the  means. 

399.   Written  Exercises. 
Find  the  missing  term. 

1.  3:44.-:5:z. 

As  the  product  of  the  extremes  is  equal  to  the  product  of  the  means, 
3  multiplied  by  x  is  equal  to  4f  multiplied  toy  5 ;  i.e.  3  x  =  4$  x  5. 
jc,  therefore,  is  equal  to    ?  x    •     This  reduces  to  ^,  or  8.  Ans. 

To  find  an  extreme,  divide  the  product  of  the  means  by 
the  other  extreme. 

2.  %:#■■  ■.*■■& 

The  product  of  the  means  |f  x  X  equals  the  product  of  the  extremes 
|  x  J£.  x  is  equal,  therefore,  to  $  x  ||  h-  |£»  Inverting  the  divisor, 
we  have  |  x  £J  x  jft.     Cancel. 

To  find  a  mean,  divide  the  product  of  the  extremes  by  the 
other  mean. 

3.  3%  +  16  =  i  +  x.  6.    ?:19::28:76. 

4.  5:7::  121 :  Xm  7.    a; :  15  :  :  4  :  f 

5.  3-*-a  =  12---20.  8.    a:£::2:7. 


316  Chapter  Six. 

9.   f:*::J:f  12.   *fftill>**|; 

10.  f:f:j«:2J>  13.    SB :  9  : :  4  :  a;. 

11.  l:|::lf:aj. 

14.  1  lb.  1  oz. :  2  lb.  4  oz. :  :  17^ :  »#. 

15.  3qt.  1  pt.-r-l  gal.  =  »^-f-80^. 

16.  4  bottles  :  x  bottles  =  6  pints  :  15  pints. 

17.  x  men  :  9  men  =  16  acres  :  36  acres. 

400.   Oral  Problems. 

1.  If  9  eggs  cost  25 /,  what  will  3  dozen  cost? 
Explanation.  —  3  dozen,  or  36,  will  cost  4  times  as  much  as  9  ;  4 

times  25  j*  =  $1. 

2.  If  7  pounds  of  flour  cost  23^,  what  will  be  paid  for 
49  pounds  ? 

3.  For  $5  1  can  get  12  straw  hats.  How  many  can  I 
get  for  $20? 

4.  A  wheel  makes  75  revolutions  in  5  minutes.  How 
many  does  it  make  in  an  hour  ? 

5.  $100  principal  gives  $6  interest.  How  much  will 
be  the  interest  of  $  450  principal  ? 

6.  A  merchant  pays  75^  freight  on  125  pounds  of  mer- 
chandise. How  much  will  be  the  freight  on  1000  pounds  at 
the  same  rate  ? 

7.  A  locomotive  goes  3  miles  in  4  minutes.  How  far 
does  it  go  in  an  hour  ? 

8.  4  horses  can  eat  a  certain  quantity  of  hay  in  10 
months.     How  long  will  it  last  20  horses  ? 

9.  12  men  can  do  a  piece  of  work  in  15  days.  How  long 
will  36  men  require  ? 

10.   15  yards  cost  270  cents.     What  will  be  the  cost  of  5 
yards  ? 


Proportion.  317 

401.   Written  Problems. 

1.  If  9  cows  cost  $267,  what  will  be  the  cost  of  36  at 
the  same  rate  ? 

The  ratio  of  the  cost,  $  267  :  $  x,  must  be  the  same  as  the  ratio  of  the 
number  of  cows,  9  :  36.     Making  the  proportion,  we  have 

9  :  36  : :  267  :  x. 

mu      t  9  267  X  36 

Therefore,  x  = - —  • 

9 

Cancelling,  x  =  $  1068.  Ans. 

2.  7  barrels  of  sugar  cost  $  104.32.  Find  the  cost  of  42 
barrels  at  the  same  rate. 

3.  A  wheel  makes  248  revolutions  in  5  minutes.  How 
many  does  it  make  in  1  hour  20  minutes  ? 

Make  the  required  number  of  revolutions  the  fourth  term.  The 
proportion  will  then  be  as  follows : 

5  minutes :  80  minutes  : :  248  revolutions  :  x  revolutions. 

_  248  revolutions  x  80 
5 

4.  A  locomotive  goes  2.8  miles  in  4  minutes.  How  far 
does  it  go  in  an  hour  ? 

5.  From  9  pounds  of  yarn  are  made  42  yards  of  dress 
goods.  How  many  yards  can  be  made  from  165  pounds  of 
yarn? 

How  many  pounds  of  yarn  are  needed  for  196  yards  of 
goods  ? 

6.  If  17  men  receive  $  357  for  a  week's  work,  how  much 
should  24  men  receive  ? 

7.  If  17  men  take  27  days  to  finish  some  work  how  long 
would  it  take  51  men  ? 

Note.  — The  work  done  by  51  men  would  be  fy  of  the  work  done 
by  17  men.  The  time  required  by  51  men  would  be  £|  of  the  time  re- 
quired by  17  men. 


3 18  Chapter  Six. 

8.  When  a  sura  of  money  is  divided  among  48  persons, 
each  receives  $  27.50.  How  much  would  each  receive  if  the 
same  sum  were  divided  among  16  persons  ? 

9.  For  $  85  I  can  purchase  238  yards  of  dress  goods. 
How  many  yards  can  I  purchase  for  $  5  ? 

10.  A  can  do  a  piece  of  work  in  6  days ;  B  can  do  it  in  7 
days.  If  B's  wages  are  $  2.10  per  day,  how  much  should 
A  receive  per  day  ? 

11.  If  for  7s.  6d.  I  can  buy  9  pounds  of  raisins,  how  many 
pounds  can  I  buy  for  £  56  16s.  ? 

12.  A  quantity  of  provisions  would  last  a  ship's  crew  20 
days,  allowing  each  man  2  lb.  4  oz.  daily.  What  should 
each  man  be  allowed  so  as  to  make  the  provisions  last  4 
days  longer  ? 

24  days :  20  days  :  :  36  ounces  :  x  ounces. 

13.  If  40  men  are  able  to  do  a  piece  of  work  in  10  hours, 
how  many  extra  men  must  be  employed  to  finish  it  in  8 
hours  ? 

8  hours  :  10  hours  : :  40  men  :  x  men.  The  number  of  extra  men  is 
x-40. 

14.  If  it  requires  40  yards  of  carpet  2  ft.  9  in.  wide  to 
cover  a  floor,  how  many  yards  of  carpet  2  ft.  6  in.  wide 
would  be  needed  ? 

15.  How  long  will  it  take  a  train  to  go  112  miles,  at  the 
rate  of  46  miles  in  1  hr.  20  min.  30  sec.  ? 

16.  If  a  beam  5  ft.  6  in.  long,  10  inches  wide,  and  8  inches 
thick  weighs  924  pounds,  find  the  length  of  another  beam 
of  the  same  material  which  weighs  3024  pounds,  and  whose 
end  is  a  square  foot. 


Partitive  Proportion.  319 

PARTITIVE  PROPORTION. 

402.  Preliminary  Exercises. 

1.  Coin  silver  consists  of  9  parts  silver  and  1  part  cop- 
per. What  is  the  ratio  of  the  weight  of  the  silver  in  a  dime 
to  the  weight  of  the  coin  ? 

2.  What  is  the  ratio  of  the  weight  of  the  copper  to  the 
weight  of  the  coin  ? 

3.  How  many  ounces  of  copper  are  there  in  a  bar  of 
coin  silver  weighing  90  ounces  ?  How  many  ounces  of  pure 
silver  ? 

403.  Partitive  proportion  is  the  process  of  dividing  a  num- 
ber into  parts  proportional  to  given  numbers. 

404.  Written  Problems. 

1.  Divide  180  into  parts  proportional  to  2,  3,  and  4. 

If  the  parts  were  2,  3,  and  4,  the  whole  number  would  be  2  + 

2  3  +  4,  or  9.     The  ratio  of  the  whole  to  the  first  part  must  be  9  to 

3  2 ;  of  the  second,  9  to  3  ;  of  the  third,  9  to  4.     These  ratios  give 

4  rise  to  the  proportions  indicated. 

9:2::180:z.  .\z  =  40. 
9:3::  180:  y.  .'.?/  =  60. 
9  :  4  : :  180  :  z.     .'.  z  =  80.     Ans.  40,  60,  and  80. 

2.  Gunpowder  is  composed  of  15  parts  of  saltpeter,  2  of 
sulphur,  and  3  of  charcoal,  mixed  together.  How  many 
pounds  of  each  are  there  in  72  pounds  of  powder  ? 

15  In  a  mixture  of  15  lb.  -f  2  lb.  +  3  lb.,  or  20  lb.,  there  will  be 

2  15  lb.  saltpeter;   hence,  the  ratio  of  the  whole  weight  to  the 

3  weight  of  the  saltpeter  is  20  lb.  to  15  lb.,  etc. 

20 :  15  :  :  72  :  number  of  pounds  of  saltpeter. 
20  :    2  : :  72  :  number  of  pounds  of  sulphur. 
20  :    3 :  :  72  :  number  of  pounds  of  charcoal. 


320  Chapter  Six. 

3.  A  bankrupt  surrenders  property  worth  $1287  for  the 
benefit  of  three  creditors  to  whom  he  owes  $750,  $1125, 
and  $  1245,  respectively.  How  much  should  each  creditor 
receive  ? 

4.  A  had  on  storage  in  a  warehouse  2400  bales  of  cotton, 
B  1500  bales,  and  C  1100  bales.  After  a  fire  that  destroyed 
all  distinguishing  marks,  the  damaged  cotton  was  sold  for 
$  10,000.     How  should  this  sum  be  divided  ? 

5.  Our  standard  gold  coin  consists  of  900  parts  gold,  90 
parts  silver,  10  parts  copper.  What  is  the  quantity  of  each 
metal  in  50  pounds  of  coin  ? 

6.  Two  men  hire  a  pasture  for  $45.  Cme  puts  in  15 
cows ;  the  other  puts  in  12  cows.     What  should  each  pay  ? 

7.  A  and  B  hire  a  boat  for  50  days,  paying  $30.  A 
uses  it  27  days;  B  uses  it  23  days.  How  much  should 
each  pay? 

8.  Three  farmers  together  paid  $54  for  threshing  their 
grain.  A  threshed  his  crop  of  900  bu. ;  B  threshed  his  crop 
of  828  bu. ;  C  672  bu.     What  did  each  pay  ? 

9.  A  and  B  contract  to  haul  a  pile  of  lumber  for  $  105.  A 
furnishes  3  teams,  and  B  4  teams.  How  much  does  B 
receive  ? 

10.  Three  merchants  shipped  a  cargo  of  iron  by  sea.  A 
gent  180  tons,  B  sent  105  tons,  C  sent  315  tons.  During  a 
storm  the  sailors  were  obliged  to  throw  overboard  180  tons 
to  save  the  vessel.  Assuming  that  the  cargo  should  sustain 
one  fourth  of  the  loss,  what  portion  of  the  loss  should  each 
merchant  sustain? 

11.  Divide  90  into  two  parts  which  shall  be  to  each  other 
as  9  to  1. 


Partnership.  321 

•  PARTNERSHIP. 

405.   Written  Problems. 

1 .  B  and  C  gain  by  trade  $  182.  What  is  the  gain  of 
each,  B  having  put  in  $300,  and  C  $400  ? 

The  total  investment  is  $ 700.     The  ratio  of  the  total  invest- 
300      ment  to  B's  investment  is  700  to  300.    This  should  be  the  ratio 
400      of  the  total  profit  to  B's  share,  etc. 
700  :  300  :  :  $  182  :  B's  share. 
700:400::$182:C's  share. 

Make  proportions  whose  antecedents  in  each  case  are  the 
total  investment  and  the  total  profit,  the  consequents  being  the 
investment  of  one  partner  and  his  share  of  the  profit. 

2.  A,  B,  and  C  invest  $  720,  $  340,  and  $  960,  respectively. 
The  profits  are  $  101.     What  is  each  one's  share  ? 

3.  A,  B,  and  C  buy  a  house  for  $7500.  A  furnishes 
$2000;  B,  $2500;  C,  the  remainder.  The  yearly  rent,  less 
expenses,  is  $  576.     To  what  amount  is  each  entitled  ? 

4.  M  and  N  entered  into  partnership.  M  puts  $  200  into 
the  business  for  5  months,  and  N  $  300  for  4  months.  They 
gained  $  132.    Find  the  share  of  each. 

An  investment  of  $  200  for  5  months  is  equivalent 
200  x5  =  1000     to  an  investment  of  $  1000  for  1  month  ;  an  invest- 
300  X  4  =  1200     ment  of  $  300  for  4  months,  to  $  1200  for  1  month. 
2200  :  1000  :  :  $  132  :  M's  share. 
2200  :  1200  : :  $  132  :  N's  share. 
In  ascertaining  the  ratio  of  the  whole  capital  to  the  share  con- 
tributed by  each,  $1000  and  $1200  are  taken  as  representing  the 
shares  of  each  in  a  total  capital  of  $2200. 

Multiply  each  partner's  share  of  the  capital  by  the  time  it  is 
in  the  busiiiess,  and  consider  the  products,  respectively,  as  the 
sums  contributed  by  the  partners. 

Note.  —  This  mode  of  ascertaining  a  partner's  share  of  profits  or 
losses  is  based  upon  the  assumption  that  the  agreement  of  the  partners 
does  not  provide  for  a  different  division. 


322  Chapter  Six. 

5.  X  and  Y  rent  a  field  for  $  32.  X  puts  in  8  horses  for 
6  months,  and  Y  10  horses  for  8  months.  How  many  dol- 
lars should  each  pay  ? 

8  horses  for  6  months  =  how  many  for  one  month  ? 
10  horses  for  8  months  =  how  many  for  one  month  ? 

6.  Three  men  hire  a  pasture  for  $  84.  One  puts  in  15 
cows  for  12  weeks;  the  second  puts  in  20  cows  for  6  weeks; 
the  third  puts  in  18  cows  for  10  weeks.  What  amount 
should  each  pay  ? 

7.  Four  men  hire  a  pasture  field  together.  The  first 
pastures  4  cows  18  weeks ;  the  second,  5  cows  12^  weeks ; 
the  third,  11  cows  6J  weeks ;  the  fourth,  9  cows  16  weeks. 
What  part  of  the  rent  should  each  pay  ? 

8.  Two  men  hire  a  pasture  for  f  420.  A  puts  in  300 
sheep  for  5  weeks,  and  B  puts  in  450  sheep  for  6  weeks. 
What  should  each  pay  ? 

9.  A,  B,  and  C  enter  into  partnership.  A  puts  in  $500 
for  4  months,  B  $400  for  6  months,  and  C  $800  for  3 
months ;  they  gain  $  340.  Find  each  man's  share  of  the 
gain. 

10.  A  partnership  is  formed  between  A  with  a  capital  of 
$  1500  and  B  with  a  capital  of  $  2500.  Six  months  there- 
after they  take  in  C  with  a  capital  of  $  4000.  How  should 
a  profit  of  $  3500  be  divided  at  the  end  of  the  year  ? 

11.  A  and  B  form  a  partnership.  A  furnishes  $  2000,  B 
$  3000.  After  a  year  A  furnishes  an  additional  $  1000. 
At  the  end  of  2  years  the  business  is  disposed  of  for  $  7100. 
How  much  should  each  receive  ? 

Suggestion.  —  A  receives  his  $3000  and  how  much  of  the  profits  ? 
Should  he  receive  as  much  as  B,  who  had  $  3000  in  the  business  the 
whole  time  ? 


20: 

:30 

18: 
4 

:27 
:    5: 

men    men 

:72  :  x 

15: 

:    3 

9: 

:10 

Compound  Proportion.  323 

COMPOUND  PROPORTION. 

406.  A  compound  proportion  is  one  in  which  either  ratio  is 
compound. 

407.  Written  Problems. 

1.  If  72  men  dig  a  ditch  20  yd.  long,  1  ft.  6  in.  broad,  4 
ft.  deep,  in  3  days  of  10  hours  each,  how  many  men  would 
be  required  to  dig  a  ditch  30  yd.  long,  2  ft.  3  in.  broad,  and 
5  ft.  deep,  in  15  days  of  9  hours  each  ? 

Since  the  number  of  men  is  required,  72  men 
is  made  the  third  term  of  the  proportion.  Con- 
sidering the  length  alone,  the  ratio  of  72  men  to 
the  required  number  would  be  equal  to  the  ratio 
of  20  feet  to  30  feet.  Considering  the  width, 
the  ratio  would  be  18  inches  to  27  inches.  Con- 
sidering the  depth,  the  ratio  would  be  4  feet  to  5  feet.  Considering 
the  number  of  days,  the  ratio  would  be  15  days  to  3  days.  Con- 
sidering  the   number   of    hours    per     men 

day,  the  ratio  would  be  9  hours  to      72  X  30  X  27  X  5  X  3  X  10 
10  hours.     Dividing  the  product  of  20  X  18  X  4  X  15  X  9 

the  means  by  the  product  of  the  ex- 
tremes, the  number  of  men  is  found  to  be  46. 

Place  the  number  required  as  the  fourth  term,  making  the 
like  number  the  third  term.  Arrange  the  couplets,  considering 
the  effect  of  each  separately  on  the  result.  Divide  the  product 
of  the  means  by  the  product  of  the  extremes. 

2.  If  45  horses  eat  1\  tons  of  hay  in  30  days,  how  many 
tons  should  last  84  horses  56  days  ? 

3.  If  4  men,  working  8  hours  per  day,  can  mow  a  meadow 
in  3  days,  how  many  men,  working  9  hours  per  day,  can  mow 
a  meadow  three  times  as  large  in  4  days  ? 

4.  If  10  men,  working  8  hours  per  day,  can  build  a  cer- 
tain wall  in  6  days,  how  many  hours  a  day  must  12  men 
work  to  build  the  same  wall  in  4  days  ? 


324  Chapter  Six. 

5.  If  108  men  can  build  a  fort  in  12£  days  of  12  J  hours 
each,  in  how  many  days  can  84  men  build  it  by  working  10£ 
hours  daily  ? 

6.  What  will  it  cost  to  transport  1000  pounds  of  mail 
matter  1000  miles,  at  $  1  per  100  pounds  per  100  miles  ? 

7.  If  12  men  can  do  a  piece  of  work  in  20  days,  what 
number  of  men  will  be  required  to  do  four  times  as  much 
work  in  a  fifth  part  of  the  time  ? 

8.  If  14  men  can  mow  168  acres  in  12  days  of  8  hr.  15 
min.  each,  how  many  acres  can  20  men  mow  in  11  days  of 
7  hr.  48  min.  each  ? 

9.  If  5  needlewomen  can  do  a  piece  of  work  in  11  days 
of  9  hours  each,  how  long  will  it  take  3  needlewomen  to  do 
two  such  pieces,  supposing  them  to  work  10J  hours  each 
day? 

10.  A  employs  a  capital  of  $2500  in  business,  and  at  the 
end  of  3  years  takes  into  partnership  B,  who  furnishes 
$4000.  Four  years  later  they  are  joined  by  C,  with  a  cap- 
ital of  $  5000.  At  the  end  of  12  years  from  the  commence- 
ment of  the  business  the  profits,  amounting  to  $  15,000,  are 
divided.     What  amount  should  each  receive  ? 

A's  money  is  in  the  business  how  many  years  ?  B's  how  many 
years  ?     C's  how  many  ? 

11.  A  and  B  rented  a  field  for  a  year  for  $175.  A  put 
in  6  horses  for  the  whole  time;  !B  put  in  5  horses  for  11 
months  and  3  horses  for  5  months.  How  much  of  the  rent 
had  each  to  pay  ? 

12.'  A  field  of  grain  was  to  be  cut  down  by  40  men  in  10 
days.  Eight  of  the  men,  however,  failed  to  come.  How 
long  did  it  take  the  others  to  do  the  work  ? 


Review.  325 

REVIEW. 
408.    Oral  Problems. 

1.  How  many  weeks  will  4-J-  tons  of  coal  last  Mrs. 
Bright,  if  she  uses  ^  of  a  ton  each  week  ? 

2.  I  can  buy  2  pairs  of  shoes  for  12  shillings.  How 
many  pairs  at  the  same  rate  can  I  buy  for  £  3  ? 

3.  If  two-thirds  of  your  age  is  8  years  and  4  months, 
how  old  are  you  ? 

4.  5  quarts  equal  what  decimal  of  a  peck  ? 

5.  What  is  the  cost  of  700  pounds  of  coal  at  $  7  a  ton  ? 

6.  How  much  would  you  pay  for  2|  yards  of  cloth  at 
37£  bayard? 

7.  In  what  time  will  f  50,  at  6%,  give  $  18  interest? 

8.  If  I  buy  an  article  for  $  75  and  sell  it  for  $  50,  what 
is  my  loss  per  cent  ? 

9.  By  selling  an  article  for  $9,  a  man  gained  25%. 
How  many  dollars  would  he  have  gained  if  he  had  sold  the 
article  at  an  advance  of  50  %  over  cost  ? 

10.  How  many  quarts  of  peanuts  in  1  bushel  and  3 
pecks  ? 

11.  What  would  be  the  cost  of  120  books  at  660  each  ? 

12.  Change  66,321  mills  to  dollars. 

13.  $120  is  i  per  cent  of  what  number  of  dollars  ? 

14.  In  what  time  will  $  50  doubie  itself  at  8%  ? 

15.  If  $  1  is  paid  for  insuring  a  piano  worth  $  500,  what 
is  the  rate  of  insurance  ? 

16.  Into  how  many  lots,  containing  f  of  an  acre  each,  can 
8  acres  be  divided  ? 

17.  A  man  lends  $  1200  at  6%,  and  1500  at  5%.  !  What 
is  the  difference  in  the  amount  of  yearly  interest  due  on 
each? 


326  Chapter  Six. 

18.  A  man  owning  §  of  a  ship  sold  §  of  his  share.  What 
part  of  the  ship  did  he  still  own  ? 

19.  How  many  rings,  each  2  pwt.  12  gr.,  can  be  made 
from  \  pound  of  gold  ? 

20.  Find  the  number  of  square  inches  on  the  surface  of 
a  block  10  inches  long  by  4  inches  wide  by  3  inches  thick. 

409.   Written  Problems, 

1.  A  traveller  walked  23\  miles  the  first  day,  3 J  miles 
more  the  second  day  than  the  first,  and  3J  miles  more  the 
third  day  than  the  second.  How  far  did  he  walk  in  the 
three  days  ? 

2.  Multiply  63.15  by  1.04;  divide  the  product  by  6.25, 
and  subtract  the  quotient  from  11 

3.  How  many  bricks,  8  inches  long  and  4  inches  wide, 
will  be  needed  to  make  a  sidewalk  20  feet  long  and  4  feet 
wide? 

4.  If  it  costs  $  10.24  to  carry  1500  pounds  356  miles,  what 
will  it  cost  to  carry  2700  pounds  890  miles  ? 

5.  A  house  rents  for  $  30  a  month,  and  the  owner  pays 
$  75  a  year  for  taxes  and  repairs.  What  is  the  value  of  the 
house,  if  his  net  profit  is  5  per  cent  per  annum  ? 

6.  A  loaned  B  a  sum  of  money  at  4£  per  cent  interest  per 
annum.  At  the  end  of  18  months  B  paid  the  debt,  principal 
and  interest,  in  all  $  1814.75.    What  was  the  sum  borrowed  ? 

7.  If  a  5-months  note  for  $  760,  dated  March  13,  is  dis- 
counted at  a  bank  May  23,  the  rate  being  7  per  cent  a  year, 
what  will  be  the  proceeds  ? 

8.  A  grocer  bought  40  gallons  of  maple  syrup  at  the  rate 
of  4  gallons  for  $  6,  and  sold  it  at  the  rate  of  5  gallons  for 
$  8.     What  was  the  whole  gain,  and  the  gain  per  cent  ? 


Review.  327 

9.  Two  pictures  were  sold  for  $  99  each.  On  one  there 
was  a  gain  of  10%  ;  on  the  other  a  loss  of  10%.  Was  there 
a  gain  or  a  loss  on  the  sale  of  both,  and  how  much  ? 

10.  New  York,  Jan.  1,  1904. 

One  year  after  date  I  promise  to  pay  J.  Edward  Swans- 
ton  Eight  Hundred  Dollars  for  value  received,  with  interest. 

$800T°oV  Kufus  L.  Scott. 

Indorsed  as  follows:  Apr.  1,  1904,  $10;  July  1,  1904, 
$35;  Nov.  1,1904,  $100. 
What  was  due  Jan.  1,  1905  ?     (Merchant's  Eule.) 

11.  What  is  the  difference  on  a  bill  of  $780,  between  a 
discount  of  40%  and  a  discount  of  35  and  5%  ? 

12.  How  many  cords  in  a  pile  of  wood  42  feet  long, 
12  feet  high,  and  8  feet  wide  ?  Find  its  cost  at  $  6.35  per 
cord. 

13.  What  principal,  on  interest  for  2  yr.  6  mo.  at  4%, 
will  gain  $850? 

14.  What  is  the  cost  of  insuring  a  house,  worth  $  25,000, 
for  |  of  its  value  at  1£%  ? 

15.  At  9fi  a  cubic  foot  what  will  be  the  cost  of  a  block 
of  stone  9  ft.  long,  4  ft.  wide,  and  5  ft.  6  in.  thick  ? 

16.  If  a  steeple  150  feet  high  casts  a  shadow  of  275  feet, 
how  long  a  shadow  will  be  cast  by  a  man  6  feet  tall,  at  the 
same  time  of  day  ? 

17.  The  tax  to  be  raised  in  a  certain  town  is  $1350. 
The  taxable  property  is  valued  at  $  108,000  What  is  the 
tax  on  one  dollar  ? 

18.  Mr.  Fox  buys  one-fifth  of  an  acre  of  land  for  $21.78. 
For  how  much  a  square  foot  must  he  sell  it  to  gain  20%  ? 


328  Chapter  Six. 

19.  What  is  the  cost  of  covering  a  floor  16£  ft.  long, 
12  ft.  wide,  with  oil-cloth  1%  yd.  wide,  at  75^  a  yard  ? 

20.  The  edges  of  a  large  cubical  box  are  5  feet  long. 
How  many  square  feet  of  paper  will  cover  the  outside  of 
the  box? 

21.  A  field  110  yards  long  and  44  yards  wide  contains  an 
acre.  What  is  the  area  of  a  field  220  yards  long  and  88 
yards  wide  ?     Of  one  440  yards  long  and  176  yards  wide  ? 

110  x  44  :  220  x  88  : :  1  acre  :  x  acres. 

22.  If  a  steel  bar  12  feet  long,  4  inches  broad,  and  2\ 
inches  thick  weighs  480  pounds,  what  is  the  weight  of 
another  steel  bar  18  feet  long,  3  inches  broad,  and  2  inches 
thick  ? 

23.  At  a  certain  hour  a  pole  6  feet  high  casts  a  shadow 
measuring  4  ft.  2  in.  Calculate  the  height  of  a  steeple 
whose  shadow  at  the  same  hour  is  104  ft.  2  in. 

24.  If  7  men  receive  $  126  for  5  weeks'  work,  how  much 
should  they  receive  for  9  weeks'  work  ? 

25.  If  76  boards,  each  14  feet  long  and  10  inches  wide, 
are  worth  $19.76,  how  much  would  50  such  boards  be 
worth  ? 

INVOLUTION. 

410.  Involution  is  finding  any  power  of  a  number. 

A  power  of  a  number  is  the  product  obtained  by  using  the 
number  a  certain  number  of  times  as  a  factor. 

2  is  the  first  power  of  2.  2  x  2,  or  4,  is  the  second  power 
of  2.     2  x  2  x  2,  or  8,  is  the  third  power  of  2,  etc. 

411.  The  second  power  of  a  number  is  called  its  square; 
the  third  power  is  called  its  cube. 


Involution.  329 

412.  The  power  of  a  number  is  indicated  by  writing  a  small 
figure,  called  an  exponent  a  little  to  the  right  of  the  upper 
part  of  a  number. 

The  square  of  2  is  written  22. 

The  cube  of  2  is  written  23. 

The  fourth  power  of  2  is  written  2*. 

52  =  25,  122  =  144. 

What  is  the  square  of  4  ?  Of  6?  Of  7?  Of  9?  Of  10? 
Of  11? 

Square  13.  15.  21.  16.  19.  142=?  172*=?  542  =  ? 
332=? 

413.  The  square  of  25  =  (20  +  5)  x  (20  +  5). 

20+5 

20+5 
Multiplying  by  20  202  +     20  x  5 

Multiplying  by  5  20  x  5   +  52 

202  +  2  (20  x  5)  +  52  =  400  +  200  +  25  =  625. 

414.  The  square  of  the  sum  of  two  numbers  is  equal  to 
the  square  of  the  first  -f-  twice  the  product  of  the  first  by 
the  second  -f-  the  square  of  the  second. 

.    132=(10  +  3)2=    102+-2(10x3)+32=? 
182  =  (10  +  8)2  =  100  +•  160  +•  64  =  ? 
272  =  (20  +  7)2  =  400  +  280  +  49  =  ? 


415.   Oral  Exercises. 

Square : 

1.   19. 

4.   26. 

7.   51. 

10.    32. 

13. 

27. 

2.   22. 

6.   31. 

8.    61. 

11.   42. 

14. 

33. 

8.   24. 

6.   41. 

9.    23. 

12.    52. 

15. 

43. 

23°  Chapter  Six. 

EVOLUTION. 

416.  Evolution  is  finding  any  root  of  a  number. 
A  root  is  one  of  the  equal  factors  of  a  number. 

The  square  root  of  a  number  is  one  of  its  two  equal  factors. 

The  square  root  of  4  is  2  j  of  9  is  3 ;  of  16  is  4 ;  of  25  is  5c 

417.  Give  the  square  root  of  36.    Of  64.    Of  81.    Of  121. 
Of  49.     Of  100.     Of  144. 

418.  The  sign  of  a  square  root  is  -y/. 

V81  =  9.         Vl2l  =  ?.        V25  =  ?.        V49  =  ?. 

SQUARE  ROOT. 

419.  Find  the  square  root  of  169. 

102  =  100.    202  =  400.    The  square  root  is  between  10  and  20 ;  it  is, 
therefore,  10  +  a  second  number. 

169  =  102  +  2  (10  x  second)  +  second2. 

169  =  100  +  20  x  second  +  second2. 

20  x  second  +  second2  =  69. 

From  this  it  appears  that  the  second  number  is  3,  since 

20  x  3  +  32  =  69. 

420.  It  may  be  shown  in  this  way : 

10  (first  number) 
169 
102  =  100 
Trial  divisor  —  twice  10  20)       69(3  second  number) 

60 
9 
S*  =     9 
Ans.  10  +  3  =  13. 


Square  Root.  331 

421.   Find  the  square  root  of  2116. 

40  (first  number) 
2116 
402  1600 


40  x  2  =  80,  trial  divisor  )  516(6  second  number) 

480 

36  =62 
Arts.  46. 

Instead  of  multiplying  the  trial  divisor  by  the  second  number,  and 
then  ascertaining  whether  the  remainder  is  the  square  of  the  second 
number,  the  second  number  is  added  to  the  trial  divisor  and  this  sum 
is  multiplied  by  the  second  number. 

In  practice,  the  work  is  shortened  by  omitting  the  ciphers. 

40  (first  number)  4    6  Ans. 

2116  21'16 

1600  16 


(2  x  40)  +  6  =  86)  516(6  second  number)  86)  516 

516  516 

First,  point  off  in  periods  of  two  figures,  commencing  at 
units.  Find  the  greatest  square  in  the  first  period,  and  place 
the  root  in  the  quotient.  Subtract  the  square  from  the 
first  period  and  bring  down  the  next  period.  Multiply  the 
quotient  figure  by  2,  and  use  it  as  a  trial  divisor.  Place 
the  second  figure  in  the  quotient  and  annex  it  also  to  the  trial 
divisor.  Multiply  the  figures  in  the  trial  divisor  by  the  second 
quotient  figure.  Bring  down  the  next  period,  and  proceed  as 
before  until  the  square  root  is  found. 

422.   "Written  Exercises. 
Extract  the  square  root : 

1.  196.  4.   1225.  7.   2809.  10.   6889. 

2.  324.  5.   1764.  8.   3721.  11.   8281. 

3.  676.  6.   1936.  9.   5184.  12.   9025. 


332  Chapter  Six. 

423.  Find  the  square  root : 

Note.  —  Extract  the  square  root  of  each  term  separately. 

1-  jfc  4.  ifr.  7.  #ft. 

3-    itt-  6-    t¥A-  9-    «H» 

Note.  —  Before  extracting  the  square  root  of  the  following,  reduce 
the  mixed  numbers  to  improper  fractions. 

10.  12£.  12.    2ff.  14.    156J. 

11.  11J.  13.    lOfH-  15.    264TV 

424.  Find  the  square  root  of  425,104 

6   5   2 
42'51'04 
12|5  651 

130|2  26  04  ^Ln«.  652. 

Jn  finding  any  figure  of  the  root  after  the  first,  we  multiply 
the  other  figure  or  figures  by  2  for  a  trial  divisor. 

425.  Find  the  square  root  of  20,857,489. 

4    5    6    7 


20'85'74'89 

8|5 

4  85 

90|6 

60  74 

912|7 

6  38  89  Ans.  4567. 

Find  the  square  root  of 

1.   64,516. 

4.   71,824. 

2.   73,441. 

5.   141,376. 

3.    18,769. 

6.   .702244. 

Square  Root.  333 


4.    74. 

5.    350. 

V£6=V£60 

1.89+ 
3.60'00 

2.8 

1 

2.60 

2.24 

3.69 

.3600 

.3321 

426.   Written  Exercises. 

Find  the  square  roots  to  two  decimal  places : 

1.    7.  2.    14.  3.    38. 

6.   Find  the  square  root  of  3.6. 

Note.  —  Commence  at  the  units  and 
point  off  two  places  to  the  right  as  well 
as  to  the  left,  annexing  a  decimal  cipher, 
if  necessary. 


7.    6.4.         8.    .121.         9.    .144.         10.    .196.         11.    .225 

APPLICATIONS  OF  SQUARE  ROOT. 

427.   Written  Problems. 

1.  How  many  inches  in  the  side  of  a  square  table  top 
containing  529  square  inches  ? 

2.  The  surface  of  a  square  piece  of  board  contains  3  sq. 
ft.  97  sq.  in.  What  is  the  length  of  one  side  in  feet  and 
mcnes  .  (Reduce  area  to  square  inches.) 

3.  How  many  rods  long  is  a  square  field  containing  90 
acres  ?  How  many  yards  of  fence  would  be  needed  to 
enclose  it  ? 

4.  Land  surveyors  use  a  measure  called  a  chain.  What 
is  its  length  in  rodsj  10  square  chains  being  equal  to  an 
acre  ?     What  is  the  length  in  feet  ? 

It  is  subdivided  into  100  "  links."  Find  the  length  of  a 
link  in  inches  and  decimal. 

5.  The  surface  of  the  six  equal  faces  of  a  cube  is  1350 
square  inches.  What  is  the  length  of  each  edge  of  the 
cube  ? 


334 


Chapter  Six. 


428.  Preliminary  Exercises. 

1.  Carefully  construct  a  right-angled  triangle,  base,  4 
inches,  perpendicular,  3  inches.     Measure  the  hypotenuse. 

Take  the  square  of  the  length  of  each  side  and  endeavor 
to  show  the  relation  between  the  square  of  the  hypotenuse 
and  the  squares  of  the  other  two  sides. 

2.  Construct  a  right-angled  triangle,  base,  3  inches,  per- 
pendicular, 1J  inches.  Measure  the  hypotenuse,  and  see  if 
the  relation  between  this  hypotenuse  and  the  other  two 
sides  of  this  triangle  is  the  same  as  that  found  in  the 
other  triangle. 

3.  A  right-angled  triangle  has  a  base  12  inches  long  ;  its 
perpendicular  is  3J  inches.  What  is  the  length  of  the  hy- 
potenuse ? 

4.  The  hypotenuse  of  a  right-angled  triangle  is  25  inches ; 
its  perpendicular  is  7  inches.     What  is  the  base  ? 

6.  The  base  of  a  right-angled  triangle  is  12  feet;  the 
hypotenuse  is  13  feet.     Find  the  perpendicular. 

429.  Draw  a  right-angled  triangle  (Fig.  1).  Upon  each 
side  construct  a  square  (Fig.  2).  From  the  upper  portion  of 
the  largest  square  (7,  cut  a  right-angled  triangle  of  the  same 


/m 

A 

B 

Fig.  2 


Fig.  8 


Fig.  4 


V^ 

B 

m/ 

</ 

C 

A 

Fig.  5 


dimensions  as  those  of  the  original  triangle  m.  Cut  another 
triangle  of  the  same  dimensions  from  the  left-hand  portion 
(Fig.  3).     Place  one  of  these  triangles  below  the  remainder 


Square  Root.  335 

of  the  square  G,  and  the  other  at  the  right,  as  in  Fig.  4,  and 
the  resulting  polygon  will  be  exactly  equal  in  surface  to  the 
two  squares  A  and  B  (Fig.  5). 

430.  The  square  described  on  the  hypotenuse  of  a  right- 
angled  triangle  is  equal  to  the  sum  of  the  squares  described 
on  the  other  two  sides. 

431.  Written  Exercises. 

Find  the  missing  side  of  each  of  the  following  ten  right- 
angled  triangles : 

1.  Base,  15;  perpendicular,  8 ;  hypotenuse,?. 

Square  of  hypotenuse  =  152  +  82 
=  225  +  64 
=  289. 
Hypotenuse  =  V289,  or  17. 

To  find  the  hypotenuse,  extract  the  square  root  of  the  sum 
of  the  squares  of  the  other  sides. 

2.  Base,  35 ;  perpendicular,  ? ;  hypotenuse,  37. 

P2  +  P2  =  H* ; 
352  +  P2  =  372 ; 
1225 +  P2  =  1369; 
P2  =  144; 
P=12. 

To  find  the  base  or  the  perpendicular,  extract  the  square  root 
of  the  square  of  the  hypotenuse  diminished  by  the  square  of  the 
given  side. 

3.  Base,     ?;  perpendicular,  15;  hypotenuse,  39. 

4.  Base,  20 ;  perpendicular,  21 ;  hypotenuse,    ?. 


2^6  Chapter  Six. 

5.  Base,  ?;  perpendicular,  45 ;  hypotenuse,  53. 

6.  Base,  56;  perpendicular.    ?;  hypotenuse,  65. 

7.  Base,  55;  perpendicular,  48;  hypotenuse,    ?. 

8.  Base,  ?;  perpendicular,  14;  hypotenuse,  50. 

9.  Base,  63;  perpendicular,    ?;  hypotenuse,  65. 

10.  Base,  112;  perpendicular,  15;  hypotenuse,    ?. 

11.  Find  the  area  in  acres  of  a  right-angled  triangle,  the 
length  of  the  sides  being  24  rods,  7  rods,  25  rods. 

12.  A  courtyard  84  feet  by  36  feet  is  to  be  paved  with 
flag-stones  measuring  6  feet  by  3  feet.  How  many  stones 
will  be  needed  ?  What  will  be  the  cost  of  the  work  at 
$  1.25  per  square  yard  ? 

13.  Find  the   length   of  the  fourth  ™  rd. 
side  of  the  following  piece  of  ground.     «g 

How  many  yards  in  the  perimeter?     5 
How  many  acres  does  it  contain  ?  100  rd. 

14.  Find  the  perimeter  in  rods  of  the 
field  shown  in  the  diagram. 

1  chain  =  66  feet. 
Note.  —  A  right  angle  contains  90  degrees. 

15.  What  is  the  length  of  the  diagonal  of  a  rectangular 
field  90  yards  wide,  120  yards  long  ? 

16.  The  dotted  line  in  the  accom- 
panying diagram  indicates  a  path  through 
the  field.  How  many  yards  are  saved 
by  taking  the  path  instead  of  following 
the  road  ? 


*o 


16  chains. 


36  chains. 


Koud. 


17.   Find  the   length    (in  rods   and  a  decimal)   of  the 
diagonal  of  a  square  40-acre  field. 


Cube  Root.  337 

CUBE  ROOT. 

432.  To  cube  a  number  is  to  employ  it  three  times  as  a 
factor. 

The  cube  of  4,  written  43,  is  4  x  4  x  4,  or  64. 
Find  the  cube  of  1,  9,  6,  3,  5,  8,  2,  7. 

To  find  the  cube  root  of  a  number  is  to  find  one  of  the 
three  equal  factors  of  the  number. 

The  cube  root  of  343,  written  ^343,  is  7. 

The  cube  of  25,  20  -f  5,  is  equal  to  the  following : 
We  have  seen  (Art.  413)  that 

(20  +  5)2  =  202  +  2x20  x5  +  52 

Multiplying  by  20  +  5 we  have 

Product  by  20  =  203  +  2  x  202  x  5  +        20  x  52 

Product  by  5   = 202  x  5  +  2  x  20  x  52  +  58 

(20  +  5)3  =  203  +  3  x  202  x  5  +  3  x  20  x  52  +  6* 
which  may  be  written  in  this  way, 

203+  [(3  x  202)  +  (3  x  20  x  5)  +  52]  x  5. 

433.  Extract  the  cube  root  of  15,625. 

"We  see  by  inspection  9(\  _±_K 

that  the  cube  root  is 
between  20  and  30 ; 
that  is,  20  +  a  second 
number.  Subtract  from 
15,625  the  cube  of  20, 
8,000.  The  remainder, 
7,625,  is  equal  to  the 
second  number  multi- 
plied by  the  sum  of  three  times  the  square  of  the  first  (1,200),  etc. 
Using  1,200  as  a  trial  devisor,  the  second  number  is  seen  to  be  6  or  less. 

Taking  5  as  the  second  number,  we  add  to  the  1,200  three  times  the 
product  of  the  first  and  second  (300),  and  the  square  of  the  second 
(25),  making  a  total  of  1,525.  Multiplying  this  sum  by  the  second 
number,  we  get  7,625,  which  is  equal  to  the  difference  between  15,625 
and  8,000.  The  second  number  is,  therefore,  5,  and  the  cube  root  of 
15,625  is  25. 


15,625 

(20)3  = 

8,000 

3  x202  = 

:  1,200 

7,625  remainder 

X  20  x  5  = 

:       300 

52  = 

25 

1,525 

7,625 

338  Chapter  Six. 


■#110,592  -#658,503 

40  +  8  8     7 

110,592  658'503 

40^        =              64,000  88=                512 

3x402        =4,800    46,592  3  x  802        =19,200    146,503 

3  x  40  x  8  =     960  3  x  80  x  7  =    1,680 

82=       64  72=        49 


5,824    46,592  20,929    146,503 

Ans.  48.  Ans.  87. 

In  the  last  example  we  point  off  three  places,  beginning  at  the  right, 
and  find  the  greatest  cube  in  the  first  period,  placing  its  cube  root  as 
the  first  figure  of  the  answer. 

434.  Find  the  cube  root  of  the  following : 

1.  2,197  6.   238,328  11.   ffffy 

2.  9,261  7.   421,875  12.   3.375 

3.  32,768  8.    551,368  13.    1£|£ 

4.  68,921  9.    m  14.    1^ 

5.  148,877  10.    4-ffi  15.   5m 

435.  Find  the  cube  root  of  9,938,375. 

When  the  root    contains  2      15 

more  than  two  figures,  con-  9'938'375 

tinue,  as  shown  in  the  accom-  g 

panying  example,  taking  for  3  X  202  =      1  200     1938 

divisor  three  times  the  square  3  v  20  X  1  =           60 

of  the  first  two  figures  con-  -.2=_              11 261 

sidered  as  tens,  plus  three — - — — - — — ; 

times  the  produet  of  the  first  »  X  210°  =  132  300        677  375 

two  figures  considered  as  tens  3  X  210  X  5  —      3  150 

by  the  third  figure,  plus  the  ^2  *%           *° 

square  of  the  third  figure.  135  475        677  375 

436.  Find  the  value  of  the  following: 

1.  ^1,442,897  3.    ^3,723,875  5.    #12.977875 

2.  </l,906,624  4.    ^/39,651,821  6.    #66.923416 


Measurements. 


339 


.\*«**f***+ 


MENSURATION. 
THE  CIRCLE. 

437.  A  circle  is  a  plane  figure  whose  boundary  is  at  all  points 
equally  distant  from  the  centre. 

The  curved  line  that  forms  the 
boundary  is  called  the  circumfer- 
ence. 

The  diameter  is  any  straight 
line  drawn  from  one  point  of  the 
circumference  to  another  and  pass- 
ing through  the  centre. 

The  radius  is  any  straight  line 
drawn  from  the  centre  to  the  cir- 
cumference. 

438.  Preliminary  Exercises. 

1.    Cut  out  of  stiff  paper  a  circle  whose  diameter  measures 
3^  inches.     Mark  a  point  A  on  the  circumference,  and  roll 


2? 

the  circle  on  a  plane  surface  along  the  line  MN.  Make  a 
mark  at  X  where  the  point  A  touches  the  line  at  the  begin- 
ning of  its  revolution,  and  at  Y  where  A  touches  the  line  at 
the  end.  Measure  the  distance  XY,  which  is  the  length  of 
the  circumference  of  the  circle. 

2.  If  the  distance  between  X  and  Y  is  11  inches,  what  is 
the  ratio  of  the  diameter  of  a  circle  to  the  circumference  ? 

3.  Draw  several  diameters  in  the  circle  you  have  cut  out. 
Measure  each.     How  do  the  diameters  compare  ? 

4.  What  is  the  ratio  between  the  diameter  of  a  circle  and 
the  radius  ? 


340  Chapter  Six. 

439.  "Written  Exercises, 

Find  the  circumference  of  a  circle  whose  diameter  is  3% 

inches. 

3|  inches  x  3.1416  =  10.9956  inches.  Ans. 

The  ratio  between  the  circumference  of  a  circle  and  its  diameter  has 
been  ascertained  to  be  3.1416. 

Process.  —  To  find  the  circumference  of  a  circle  multiply  the 
diameter  by  3.1416. 

1.  Find  the  circumference  of  a  circle  whose  diameter  is 
25  feet. 

2.  The  circumference  of  a  circle  measures  39.27  feet. 
What  is  the  diameter  ? 

3.  The  radius  of  a  circle  is  8  yards.  Find  its  circumfer- 
ence. 

4.  The  diameter  of  a  bicycle  wheel  is  28  inches.  How 
far  does  a  bicycle  go  during  10  revolutions  of  the  wheel  ? 

AREAS  OF  CIRCLES. 

440.  Preliminary  Exercises, 

1.  Divide  a  circle  whose  diameter  measures  3J  inches  into 
sixteen  equal  parts  by  cuts  passing  through  the  centre  in 
each  case.     Arrange  the  parts  as  shown  in  the  accompanying 

figure : 

B O 


2.   When  the  diameter  measures  3J  inches,  what  is  the 
length  of  AB  ?    What  part  of  the  diameter  is  AB  ? 


Measurements.  341 

3.  What  part  of  the  circumference  is  embraced  between 
A  and  D  ? 

4.  When  the  given  circle  is  divided  into  a  very  great 
number  of  parts,  what  will  be  the  length  of  AD  ?     Of  AB  ? 

5.  When  the  number  of  parts  is  extremely  great,  ABCD 
becomes  a  rectangle.     Find  its  area. 

The  number  of  square  inches  (feet,  etc.)  in  the  area  of  a  circle  is 
obtained  by  multiplying  one-half  the  number  of  inches  (feet,  etc.)  in 
the  diameter  by  one-half  the  number  in  the  circumference. 

This  may  be  expressed  by  the  formula  : 

Area  of  circle  =  \  diameter  x  £  circumference. 

As  the  circumference  equals  diameter  x  3.1416,  £  circumference 
equals  radius  x  3.1416.     Multiplying  by  £  diameter,  or  radius,  we 

Area  of  circle  =  square  of  radius  x  3.1416. 

To  find  the  area  of  a  circle  multiply  the  square  of  the  radius 
by  8.1416. 

441.   "Written  Exercises. 

1.  What  is  the  area  of  a  circle  whose  radius  is  36  feet? 
1  sq.  ft.  x  362  x  3.1416  =  1296  sq.  ft.  x  3.1416. 

2.  Find  the  area  of  a  circle  whose  diameter  is  50  yards. 

3.  What  is  the  area  of  a  circle  whose  circumference  is 

10  feet?                              1Q  5 

Diameter  = ;   Radius  = 


3.1416  3.1416 

B*  = 5_x_5 ;   J?2X  3,1416=  5x5x3.1416      ^^ 

3.1416x3.1416'  3.1416x3.1416 

4.  Calculate  the  area  of  a  circle  whose  radius  is  1  inch. 
Of  a  circle  whose  radius  is  2  inches.  What  is  the  ratio  of 
the  two  areas  ? 


34* 


Chapter  Six. 


5.  What  is  the  ratio  between  the  area  of  a  circle  whose 
radius  is  1  inch  and  that  of  a  circle  whose  radius  is  3 
inches  ? 

Indicate  operations  and  cancel. 

6.  How  many  square  yards  are  there  in  a  circular  walk, 
the  radius,  AB,  of  the  inner  edge  of  walk 
being  10  feet,  and  that  of  the  outer  edge, 
AC,  being  15  feet  ? 

Find  the  difference  between  the  area  of  a  circle 
of  15  feet  radius,  and  that  of  a  circle  of  10  feet 
radius. 

7.  A  circular  flower  bed  20  feet  in  diameter  is  surrounded 
by  a  walk  5  feet  wide.  How  many  square  feet  of  surface 
does  the  walk  contain  ? 

If  you  have  to  subtract  100  times  3.1416  from  225  times  3.1416,  how 
can  you  shorten  the  work  ? 

8.  How  many  square  inches  are  there  in  the  surface  of  a 
frame  3  inches  wide,  around  a  looking-glass 
6  inches  in  diameter  ? 

Area  =  ?  x  3.1416. 

9.  What  is  the  ratio  between  the  sur- 
face of  the  above  frame  and  that  of  the 
looking-glass  ? 

Indicate  operations  and  cancel. 

10.  What  is  the  radius  of  a  circle  whose  area  is  153.9384 
square  yards  ? 

11.  Find  the  radius  of  a  circle  whose  area  is  314.16  square 
inches. 

12.  Find  the  area  of  a  circle  whose  circumference  is 
15.708  feet. 


Measurements.  343 

AREAS  OF  TRIANGLES. 

442.   Written  Problems. 

1.   What  is  the  area  of  a  triangle  whose  sides  measure  15, 
16,  and  17  inches,  respectively  ? 


15 
16 


From  the  half  sum  of  the  three  sides  subtract 
each  side  separately.    The  square  root  of  the  con- 
17  tinued  product  of  the  half  sum  and  the  three 

2)48  remainders  will  be  the  area. 

9J.  _  1  K  _  Q  

24  _  16  Z  s         V24  x  9  x  8  x  7  =  Vl2^96  =  109.98 
24  —  17  =  7  Ans.   109.98  square  inches. 

2.  Find  the  area  in  square  feet  of  a  triangle  whose  sides 
measure  35  feet,  84  feet,  91  feet. 

3.  Find  the  area  of  a  triangle  whose  sides  measure  21,  28, 
and  35  rods,  respectively. 

4.  In  the  following  field,  AB  meas- 
ures 39  rods;  BC,  52  rods;  CD,  25 
rods  ;  AD,  60  rods  ;  and  the  diagonal, 
AC,  65  rods.  Find  the  area  of  the 
field  in  square  rods. 

Find  the  area  of  each  triangle  separately. 

5.  Find  the  area  of  an  isosceles  triangle  whose  base  is  30 
yards,  its  equal  sides  measuring  25  yards. 

6.  What  is  the  altitude  of  an  isosceles  triangle,  base,  64 
feet,  equal  sides,  68  feet  ?     Find  its  area. 

7.  Find  the  area  of   an  equilateral  triangle,  each  side 
being  6  feet. 

8.  Find  the  area  of  a  right-angled  triangle,  base,  42  feet, 
hypotenuse,  70  feet. 

First  ascertain  the  length  of  the  perpendicular. ' 

9.  Find  the  area  of  an  isosceles  triangle,  altitude,  48  feet, 
equal  sides,  50  feet. 


344 


Chapter  Six. 


AREAS   OF  QUADRILATERALS. 

443.   Written  Problems. 

1.  Find  the  area  of  a  square  whose  diagonal  is  150  rods. 
Suggestion.  —  Calling  one  side  of  the  square  S,  we  have  S2  +  S* 

=  1502,  150  being  the  hypotenuse  of  an  isosceles  right-angled  triangle, 
the  other  sides  being  the  sides  of  the  square.  S2  is  the  required  area. 
Do  not  find  the  length  of  S. 

2.  Find  the  area  of  the  rhomboid  (Fig.  1). 

Note.  —  The  altitude  is  the  perpendicular  of  a  right-angled  triangle 
having  a  base  of  7  rods  and  a  hypotenuse  of  25  rods. 

3.  Of  the  rectangle  (Fig.  2). 

40  rd.  80  yd. 

\     h 


-s5 


Fig.  1. 


Fig.  2. 


4.  Of  the  rhombus  (Fig.  3). 

5.  Of  the  trapezoid  (Fig.  4). 


25  rd. 


CO  rd. 


60 


40 


Fig.  3. 


Fig.  4. 


6.  Of  the  trapezium  (Fig.  5). 

7.  Of  the  rhombus  (Fig.  6). 

30  yd. 


Quadrilaterals.  345 

8.  Find  the  altitude  AB  of  the  preceding  triangle  (Fig.  7). 

(First  find  the  area.) 

9.  Find  the  diagonal  (in  rods)  of  the  square  whose  area 
is  5  acres. 

10.  Find  the  area  of  a  regular  hexagon,  composed  of  six 
equilateral  triangles,  each  side  being  6  inches  (Fig.  8). 

Note.  — A  plane  figure  bounded  by  straight  lines  is  called  a  poly- 
gon. A  three-sided  polygon  is  called  a  triangle.  A  four-sided  polygon 
is  called  a  quadrilateral.  A  hexagon  is  a  six-sided  polygon.  A  reg- 
ular polygon  is  one  having  all  its  sides  and  all  its  angles  equal.  The 
square  is  the  only  regular  polygon  of  four  sides. 


Fig.  9.  Fig.  10. 

11.  What  is  the  area  of  the  circle  circumscribed  about  the 
above  hexagon  (Fig.  9)  ? 

12.  What  is  the  area  of  the  square  inscribed  in  a  circle 
whose  diameter  is  10  feet  (Fig.  10)  ? 

13.  Find  the  area  of  a  square  circumscribing  a  circle 
whose  diameter  is  10  feet,  and  give  the  ratio  of  its  area  to 
that  of  the  inscribed  square. 

14.  Find  the  perimeter  of  a  rectangle  80  yards  long,  the 
diagonal  of  which  measures  100  yards. 

15.  A  square  piece  of  ground  containing  40  acres  is 
divided  into  4  square  fields  of  10  acres  each.  How  many 
rods  of  fence  will  be  needed  to  enclose  all  the  fields  ? 

16.  The  area  of  a  triangular  plot  is  480  square  yards. 
Two  of  the  sides  are  equal  in  length,  and  the  third  measures 
32  yards.     Find  the  perimeter. 


346 


Chapter  Six. 


SURFACES   OF  PRISMS   AND   OF  CYLINDERS. 

Note.  —  The  pupils  should  first  examine  a  number  of  prisms,  right 
and  oblique,  regular  and  irregular,  triangular,  quadrangular,  pentago- 
nal, etc.     Right  and  oblique  cylinders  should  also  be  at  hand. 

444.  A  prism  is  a  body  bounded  by  plane  faces,  two  of  which 
are  equal  and  parallel  polygons,  the  remaining  faces  being 
parallelograms. 

A  C         H  Q 


E 


F 

Fig.  1. 


s^^> 


Fig.  3. 


The  two  parallel  faces  of  a  prism  are  called  its  bases.  The 
remaining  faces  taken  together  constitute  its  convex  surface. 

In  Fig.  1,  ABC  and  DEF  are  the  bases ;  in  Fig.  2  the  bases  are 
GHIJ  and  KLMN\  in  Fig.  3,  OPQB 8  and  TUVWX. 

The  sides  AB,  CE,  etc.,  GH,  IN,  etc.,  QB,  OT,  etc.,  are  called 


445.  Prisms  may  be  either  right  or  oblique.  The  convex 
surface  of  a  right  prism  consists  of  rectangles. 

Fig.  1  is  a  right  prism  ;  Fig.  2  is  an  oblique  prism. 

Note.  —  When  a  prism  is  spoken  of,  a  right  prism  is  meant  unless 
the  word  oblique  is  used. 

The  altitude  of  a  prism  is  the  perpendicular  distance  be- 
tween the  bases. 

AD,  BF,  or  CE  is  the  altitude  in  Fig.  1.  GYis  the  alti- 
tude in  Fig.  2. 

446.  The  number  of  sides  in  each  base  determines  the 
name  as  triangular  (Fig.  1),  quadrangular  (Fig.  2),  pentagonal 
(Fig.  3),  etc. 


Surfaces. 


347 


A  quadrangular  prism  whose 
"bases  are  parallelograms  is  called 
a  parallelopipedon.  Fig.  4  is  an 
oblique  parallelopipedon.  Fig.  5 
is  a  right  parallelopipedon.  Any 
two  opposite  faces  of  a  parallelopi- 
pedon may  be  considered  the  bases. 


k\ 

1 

1 
1 

\[_ 

\ 

Fig.  4. 


Fig.  5. 


447.  When  the  bases  are  regular  polygons,  the  prism  is 
said  to  be  regular. 

Fig.  1  is  a  right  regular  triangular  prism  j  Fig.  2  is  an  oblique  irreg- 
ular quadrangular  prism. 

448.  A  cylinder  is  a  body  having  two  circular  parallel  plane 
faces,  and  one  curved  face. 


(TZD 


The  plane  faces  are  the  bases, 
face  constitutes  the  convex 


The  curved 


Fig.  6. 

surface. 

Cylinders,  like  prisms,  are  either  right 
or  oblique.  The  altitude  of  a  cylinder  is 
the  perpendicular  distance  between  the 


& 


Fig.  7. 


£2 

Fig.  & 


449.   Written  Problems. 

Note.  — The  pupils  should  be  encouraged  to  make 
cardboard  models  of  the  forms  studied. 

1.  Find  the  convex  surface  of  a  square 
prism,  one  side  of  its  base  being  4  inches  and 
its  height  6  inches.     Draw  the  development. 

Note.  — The  convex  surf  ace  is  the  surface  exclusive 
of  the  bases. 

2.  Find  the  convex  surface  of  a  triangular 
prism,  each  side  of  whose  base  measures  4 
inches  and  whose  altitude  is  6  inches.  Draw 
the  development. 


348  Chapter  Six. 

3 .  Find  the  convex  surface  of  an  hexagonal      -=^^^ 
prism,  each  side  of  its  base  being  4  inches  and 

its  altitude  6  inches.     Draw  the  development. 

4.  Can  you  show  that  the  convex  surface 

of  a  prism  is  found  by  multiplying  the  perim-       |||H  j^J 
eter  of  the  base  by  the  altitude  (height)  ? 

5.  Find  the  convex  surface  of  a  cylinder,        |j|     ||| 
the  diameter  of  its  base  being  4  inches  and  its 

height  6  inches. 

To  find  the  convex  surface  of  a  right  prism 
(or  cylinder)  multiply  the  perimeter  (circumfer-       'llillljjj^^ 
ence)  of  the  base  by  the  height. 

6.  How  do  you  find  the  entire  surface  of  a  prism  or 
cylinder  ? 

Note.  —  The  entire  surface  is  the  surface  including  the  bases. 

7.  What  is  the  entire  surface  of  a  cube  whose  side  is 
7  inches  ?     Of  a  cube  whose  side  is  12  inches  ? 

8.  The  entire  surface  of  a  cube  is  216  square  inches. 
What  is  the  length  of  one  side  ? 

Suggestion.  —  Calling  the  length  of  one  side  L,  the  area  of  each 
face  will  be  £2,  and  of  the  six  faces,  6  L2.    Then,  6  L2  =  216. 

9.  The  convex  surface  of  a  cube  is  144  square  inches. 
Find  the  entire  surface. 

How  many  faces  in  the  convex  surface  ? 

10.  Find  the  entire  surface  of  a  square  prism,  one  side  of 
whose  base  measures  4  inches,  and  whose  altitude  is  6  inches. 

Entire  surface  =  convex  surfaces  +  areas  of  bases. 

11.  The  convex  surface  of  a  square  prism  is  600  square 
feet,  the  altitude  is  15  feet.  What  is  the  length  of  one  side 
of  the  base  ? 

12.  The  entire  surface  of  a  square  prism  is  1650  square 
inches.  One  side  of  the  base  measures  15  inches.  What  is 
its  convex  surface  ?     What  is  its  altitude  ? 

Convex  surface  =  entire  surface  —  area  of  bases.  • 


Surfaces. 


349 


SURFACES  OF  PYRAMIDS  AND  CONES. 

450.  A  pyramid  is  a  body  whose 
convex  surface  is  made  up  of  triangles 
having  a  common  vertex,  the  base  of  the 
pyramid  being  a  polygon. 

Pyramids  are  either  right  or  oblique; 
regular  or  irregular  ;  triangular,  quadrangular,  pentagonal,  etc. 

In  a  right  pyramid,  each  of  the  triangles  that  make  up 
the  convex  surface  is  isosceles.    When, 
in  addition,  the  pyramid  is  a  regular 
one,  these  triangles  will  be  equal  to 
each  other. 

The  altitude  of  any  of  these  equal 
triangles  constitutes  the  slant  height  of 
a  right  regular  pyramid.  The  altitude  of 
the  prism  is  measured  by  a  line  drawn 
from  the  apex  to  the  centre  of  the  base. 

AG  is  the  slant  height  of  the  square  pyramid,  Fig.  3.     AF  is  its 
altitude. 

451.  The  cone  is  a  body  having 
a  single  circular  base,  and  a  curved 
convex  surface  sloping  to  the  apex. 

In  the  right  cone,  Fig.  4,  HI  is  the 
)j  slant  height,  and  HK  is  the  altitude. 
LO  is  the  altitude  of  the  oblique  cone, 
Fig.  4.       Fig>  5# 


Fig.  3. 


452.   Written  Problems. 

1.  The  convex  surface  of  a  square  pyramid 
consists  of  how  many  equal  triangles  ?  Find 
the  convex  surface  when  one  side  of  its  base 
measures  4  inches  and  its  slant  height  (AX) 
6  inches. 

2.  Draw  the  development. 


35° 


Chapter  Six. 


To  find  the  convex  surface  of  a  pyramid  (or  cone)  multiply 
the  perimeter  (circumference)  of  the  base  by  one-half  the  slant 
height. 

3.  Find  the  entire  surface  of  the  above  pyramid. 
Entire  surface  =  convex  surface  +  area  of  base. 

4.  Calculate  the  entire  surface  of  a  square  pyramid  whose 
slant  height  is  18  inches,  the  area  of  its  base  being  144 
square  inches. 

5.  Draw  the  developed  convex  surface  of  a 
cone,  the  diameter  of  whose  base  is  4  inches, 
and  whose  slant  height  is  6  inches. 

Calculate  the  convex  surface. 

6.  How  many  square  inches  of  paper  would 
be  required  to  cover  the  side  and  the  base  of  a  cone  6  inches 
in  diameter  at  the  base,  and  having  a  slant  height  of  10 
inches  ? 

7.  Calculate  the  slant  height  of  a  cone  whose 
altitude  is  12  inches,  the  diameter  of  its  base 
being  10  inches.  What  is  its  convex  surface  ? 

Note. — The  slant  height  is  the  hypotenuse  of  a 
right-angled  triangle,  the  other  sides  measuring  12  in. 
and  5  in.,  respectively. 

8.  What  is  the  convex  surface  of  a  cone, 
the  diameter  of  whose  base  is  6  inches,  and  its  slant  height 
10  inches  ?     Draw  the  development. 

6  in.  9.   A       semicircular 

piece  of  paper  6  inches 

in    diameter   is  folded 

into     a     hollow     cone 

(without  overlapping). 
What  will  be  the  diameter  AB  of  the  mouth 
of  the  cone  (the  base)  ?    What  will  be  the  slant  height  BC? 


Volumes.  351 


VOLUMES  OF  PRISMS  AND  OF  PYRAMIDS;  OF  CYLINDERS 
AND  OF  CONES. 

453.  Written  Problems. 

Suggestion.  —  Have  the  pupils  construct  of  cardboard  a  hollow 
square  prism  of  convenient  size,  and  a  pyramid  having  base  and  alti- 
tude respectively  equal  to  those  of  the  prism.  Let  them  use  sand  or 
water  to  ascertain  how  many  times  the  contents  of  the  pyramid  must 
be  taken  to  exactly  fill  the  prism. 

Volume  of  prism  or  cylinder  =  area  of  base  x  altitude. 

Volume  of  pyramid  or  cone  =  area  of  base  x  \  altitude. 

1.  Find  the  volume  of  a  square  pyramid,  the  area  of  the 
base  being  9  square  feet  and  the  altitude  6  feet. 

1  cu.  ft.  x  9  x  i  of  6. 

2.  What  is  the  volume  of  a  square  pyramid  whose  alti- 
tude is  12  inches,  one  side  of  the  base  being  10  inches  ? 

3.  The  base  of  a  prism  is  a  triangle  whose  sides  measure 
3,  4,  and  5  inches  respectively.  Find  the  solidity,  its  alti- 
tude being  10  inches. 

4.  The  base  of  a  prism  19  feet  high  is  a  rectangle  whose 
sides  are  9  feet  and  13  feet.  How  many  cubic  yards  does  it 
contain  ? 

5.  Find  the  volume  of  a  prism  whose  bases  are  equi- 
lateral triangles,  each  side  being  4  feet,  and  the  height  of 
the  prism  being  12  feet. 

6.  How    many    cubic    feet    are  6  ft. 
there  in  a  stone  roller  6  feet  long, 
8  feet  in  circumference  ? 

7.  Find  the  volume  of  a  cone 
whose  altitude  is  18  meters,  diame- 
ter of  base  6  meters. 


35* 


Chapter  Six. 


SURFACE  OF  SPHERE. 

454.  A  sphere  is  a  body  all 
points  on  whose  surface  are  equally 
distant  from  the  centre. 

The  distance  from  the  centre 

to   the    surface   is    called    the    Ffc— 

radius  of  the  sphere.  The  di- 
ameter is  a  line  running  between 
two  points  on  the  surface  and 
passing  through  the  centre. 

CG}  CK,  CD,  CF,  and  CJ  are  radii ;  AD  and  FG  are  diameters. 

455.  If  a  sphere  be  cut  through  at  any  part,  the  cut  sur- 
face will  be  a  circle.  When  the  cutting  plane  passes 
through  the  centre  of  the  sphere,  the  circle  is  called  a  great 
circle;  other  circles  are  called  small  circles. 

FXGG  is  a  great  circle  ;  HYIB  and  JLEZ  are  small  circles. 


A 


456.  Take  a  wooden  hemisphere  and  drive  a  tack  into  the 
centre  of  its  curved  surface.  Commencing  at  the  tack,  care- 
fully wind  a  waxed  cord  about  the  curved  surface,  in  the 
way  a  boy  winds  a  top.  When  this  surface  is  exactly  cov- 
ered, cut  the  cord. 


Wind  the  same  cord  around  a  tack  driven  into  the  plane 
surface  of  the  base  of  the  hemisphere,  pressing  it  closely  to 
the  surface.  When  the  latter  is  entirely  covered,  just  one- 
half  of  the  cord  will  be  used. 


The  Sphere.  353 

As  a  hemisphere  is  made  by  passing  the  cutting  plane 
through  the  centre  of  the  sphere,  its  base  is  a  great  circle 
of  the  sphere. 

The  above  experiment  shows  that  the  surface  of  the  hemi- 
sphere is  equal  to  that  of  two  great  circles  of  the  same  sphere. 

457.  The  surface  of  a  sphere  is  equal  to  that  of  four  great 
circles. 

Since  the  surface  of  a  great  circle  of  the  sphere  is  R2  x 
3.1416,  the  surface  of  the  sphere  is  4  R2  x  3.1416  =  D2  x 
3.1416. 

To  find  the  surface  of  a  sphere,  multiply  the  square  of  the 
diameter  by  S.lJf.16. 

458.  Written  Problems. 

1.  Find  the  surface  of  a  sphere  whose  radius  is  1  inch. 

2.  The  diameter  of  a  sphere  is  2  inches.    Find  its  surface, 

3.  What  is  the  surface  of  a  sphere  whose  circumference 
is  6.2832  inches  ? 

4.  At  10  cents  a  square  foot,  what  will  be  the  cost  of  gild- 
ing a  sphere  12  inches  in  diameter  ? 

5.  Find  the  ratio  between  the  surface  of  a  sphere  1  foot 
in  diameter,  and  the  convex  surface  of  a  cylinder  1  foot  high, 
the  diameter  of  the  base  1  foot. 

6.  What  is  the  ratio  between  the  surface  of  a  sphere 
1  foot  in  diameter,  and  the  entire  surface  of  a  cylinder 
1  foot  high,  the  diameter  of  the  base  1  foot  ? 

7.  Find  the  surface  of  a  sphere  whose  circumference  is 
20  inches. 

8.  What  is  the  ratio  between  the  surfaces  of  two  spheres 
whose  diameters  are  1  inch  and  2  inches,  respectively  ? 

9.  Find  the  ratio  between  the  surfaces  of  two  spheres 
whose  diameters  are  2  feet  and  13  feet,  respectively. 


354 


Chapter  Six. 


VOLUME  OF  SPHERE. 

459.   Cut  up  a  sphere  (a  round  potato,  for  instance)  into  a  number 
of  small  pieces,  passing  the  knife  in  each  case  through  the  centre  of  the 


sphere.    Each  piece  is  a  solid,  having  for  its  base  a  portion  of  the  sur- 
face of  the  sphere,  and  for  its  altitude  the  radius  of  the  sphere. 

When  the  pieces  become  very  numerous,  the  base  of  each  may  be 
considered  a  plane  and  the  solid  a  pyramid.      The  volume  of  each 


pyramid  is  equal  to  the  base  x  |  altitude  ;  and  the  total  volume  of  all, 
which  is  the  volume  of  the  sphere,  is  equal  to  the  total  surface  of  all 
the  bases,  which  is  the  surface  of  the  sphere,  multiplied  by  |  altitude, 
that  is,  |  radius,  or  £  diameter. 

Surface  of  sphere  =  D2  x  3.1416, 
therefore,  volume  of  sphere  =  D2  x  3.1416  x  £  D  = 

^X  3.1416. 

To  find  the  volume  of  a  sphere,  multiply  one-sixth  of  the  cube 
of  the  diameter  by  8.1J/.16. 

460.   Written  Problems. 

1.  Find  the  volume  of  a  sphere  whose  radius  is  3  inches. 

1  cu.  in  x  36  x  3.1416. 

2.  If  the  diameter  of  a  sphere  is  3  inches,  what  is  its 
volume  ? 


The  Sphere.  355 

3.  What  is  the  ratio  between  the  volumes  of  two  spheres 
whose  diameters  are  one  foot  and  two  feet,  respectively  ? 

4.  Find  the  ratio  between  the  volume  of  a  sphere  1  foot 
in  diameter,  and  that  of  a  cube  whose  side  is  1  foot. 

5.  The  radius  of  a  sphere  is  18  inches.  What  is  the  cir- 
cumference of  a  great  circle  ?     The  surface  ?     The  volume  ? 

6.  What  is  the  weight  of  an  iron  cannon-ball  12  inches  in 
diameter,  considering  the  weight  of  a  cubic  foot  of  water  as 
1000  ounces,  and  considering  iron  7.5  times  as  heavy  as 
water  ? 

7.  Find  the  ratio  between  the  volume  of  a  sphere  4  inches 
in  diameter,  and  that  of  a  cylinder  4  inches  in  altitude, 
diameter  of  base  4  inches. 

Note.  —  Indicate  the  volume  of  each,  and  cancel. 

8.  A  man  has  a  cubical  block  of  hard  wood,  its  side 
measuring  one  foot,  which  he  wishes  made  into  a  sphere  one 
foot  in  diameter.  What  decimal  part  of  the  block  is  cut 
away? 

The  volume  of  the  sphere  is  about  what  fraction  of  the 
volume  of  the  cube  ? 

MISCELLANEOUS. 

461.   Written  Problems. 

1.  If  a  piece  of  cloth  is  20  yards  long  and  f  yards  broad, 
how  broad  is  another  piece  of  cloth  12  yards  long  that  con- 
tains as  many  square  yards  as  the  former  ? 

2.  An  iron  beam  16  feet  long,  2\  feet  broad,  and  8  inches 
thick,  weighs  1280  pounds.  What  is  the  length  of  a  similar 
beam  whose  breadth  is  Z\  feet,  thickness  7£  inches,  and 
weight  2028  pounds  ? 

3.  What  will  it  cost  to  carpet  a  room  22^  feet  long  by 
15|  feet  wide  with  carpet  2 \  feet  wide,  costing  $1.50  per 
yard? 


3S&  Chapter  Six. 

4.  What  is  the  length  of  a  box  6|  feet  wide  and  1\  feet 
high,  that  will  exactly  contain  12  boxes  4£  feet  long,  2>\  feet 
wide,  and  2\  feet  deep  ? 

5.  What  is  the  value,  at  $120  per  acre,  of  a  square  field 
whose  side  is  35.25  chains? 

10  square  chains  =  1  acre. 

6.  Find  the  capacity,  in  bushels,  of  a  bin  22  feet  long, 
14  feet  wide,  12  feet  high  ? 

7.  How  many  gallons  will  a  tank  hold,  its  dimensions 
being  4  ft.  1  in.  by  3  ft.  8  in.  by  2  ft.  3  in.  ? 

8.  How  many  square  yards  are  there  in  the  walls  and 
the  ceiling  of  a  room  21  feet  long,  18  feet  wide,  12  feet 
high? 

9.  A  tank  5%  feet  by  6  feet  by  7  feet  can  be  emptied  by 
two  pipes,  one  of  which  discharges  9  gallons  per  minute  and 
the  other  7  gallons  per  minute.  How  long  will  it  take 
each  to  empty  the  tank  ?  How  long  will  it  take  both 
together  ? 

10.  A  parlor  is  18  feet  long,  15  feet  wide.  Make  a  dia- 
gram, showing  how  carpet  27  inches  wide  can  be  laid  without 
cutting  the  carpet  lengthwise.  Which  would  be  the  better 
way  to  lay  carpet  30  inches  wide  in  the  above  room  ? 

11.  Calculate  the  number  of  running  yards  of  carpet  30 
inches  wide  needed  for  the  floor  of  the  above  room,  including 
4 i  yards  wasted  in  matching  the  pattern. 

Find  the  cost  of  carpeting  the  room  at  95  cents  per 
running  yard  for  carpet,  5  cents  per  square  yard  for  lining, 
and  10  cents  per  running  yard  for  sewing  and  laying. 

12.  A  room  is  18  feet  wide,  24  feet  long,  and  9  feet  high. 
There  are  two  doors  4  feet  wide,  7-J-  feet  high ;  two  windows 
4  feet  wide,  6  feet  high ;  and  a  fireplace  5  feet  square.  How 
many  square  feet  of  plastering  will  there  be  on  the  walls 


Miscellaneous.  357 

and  ceiling,  deducting  for  a  baseboard  12  inches  wide  ?    How 
many  running  feet  of  baseboard  will  be  needed  ? 

Draw  "development"  of  the  above  room,  showing  the  four 
walls  and  the  ceiling,  and  locating  the  doors,  the  windows, 
and  the  baseboard. 

Do  not  use  baseboard  where  it  is  not  required. 

13.  At  the  rate  of  $1400  for  a  pile  of  lumber  25  feet 
long,  20  feet  wide,  10  feet  high,  what  is  the  value  of  a  pile 
50  feet  long,  40  feet  wide,  20  feet  high  ? 

14.  If  it  costs  $  14  to  paint  the  walls  and  the  ceiling  of  a 
room  25  feet  long,  20  feet  wide,  and  10  feet  high,  what  will 
it  cost  to  paint  the  walls  and  the  ceiling  of  a  room  50  feet 
long,  40  feet  wide,  and  20  feet  high  ? 

15.  Measure  accurately  the  interior  dimensions  of  a  quart 
or  a  pint  cup,  and  calculate  its  volume. 

Note.  —  How  many  cubic  inches  in  a  quart,  liquid  measure  ? 

462.   Circular  Measure. 

60  seconds  (")       1  minute  (*). 
60  minutes  1  degree  (°). 

360  degrees  1  circle. 

16.  If  the  equatorial  circumference  of  the  earth  is  25,000 
miles,  how  many  miles  apart  are  two  places  on  the  equator, 
the  distance  between  them  being  20°  ? 

20°  =  J;  circle. 

17.  What  is  the  length  of  a  degree  on  a  circle  whose 
diameter  is  18  feet  ? 

18.  The  60th  parallel  of  latitude  is  a  circle  one-half  as 
long  as  the  equator.  How  many  miles  due  east  of  Christi- 
ania  is  St.  Petersburg,  both  situated  on  this  parallel,  the 
former  being  10°  east  of  Greenwich,  and  the  latter  30° 
east? 


358  Chapter  Six. 

LONGITUDE   AND   SOLAR  TIME. 

Note.  —  This  topic  should  be  taught  in  connection  with  the  study 
of  Mathematical  Geography.  The  globe  should  be  used  to  show  the 
pupils  that  all  places  on  the  same  meridian  have  the  same  solar  time, 
that  a  difference  in  longitude  of  15  degrees  produces  a  difference  in 
time  of  1  hour,  and  that  the  more  easterly  of  two  places  has  the  later 
time. 

463.   Preliminary  Exercises. 

1.  The  difference  in  time  being  1  hour  for  each  15  degrees, 
find  the  difference  in  longitude  between  two  cities  differing 
in  solar  time  3  hours. 

2.  Two  places  differ  in  longitude  60  degrees.  What  is 
their  difference  in  solar  time  ? 

3.  London  is  75°  east  of  Philadelphia.  When  it  is  1 
o'clock  at  Philadelphia,  what  is  the  time  at  London  ? 

4.  When  it  is  2  p.m.  at  London,  what  is  the  time  at  Phila- 
delphia ? 

5.  How  many  degrees  of  longitude  correspond  to  a 
difference  of  3  hr.,  40  min.  in  solar  time? 

6.  What  is  the  difference  in  longitude  between  Phila- 
delphia, 75°  west  longitude,  and  St.  Petersburg,  30°  east 
longitude  ? 

7.  Washington  is  in  77°  west  longitude,  and  uses  "stand- 
ard time,"  that  is,  the  time  of  75°  west  longitude.  What 
is  the  difference  between  the  correct  time  at  Washington 
and  its  clock  time  ? 

8.  A  town  in  84°  west  longitude  uses  standard  time, 
that  of  90.  What  is  the  correct  time  when  the  clocks  are 
striking  12,  noon  ? 

9.  Chicago  is  87°  35'  west  of  Greenwich.  Is  it  earlier 
or  later  than  noon  at  Chicago  when  it  is  noon  at  Greenwich  ? 
Why? 


Standard  Time. 


359 


STANDARD  TIME. 

464.  In  1883,  the  railroads  of  the  United  States  adopted 
a  system  of  dividing  the  country  into  four  time  sections, 
each  of  15°  longitude.  The  75th  meridian  west  of  Green- 
wich, which  passes  between  New  York  and  Philadelphia, 
was  selected  as  the  starting-point.  The  section  governed 
by  the  time  of  this  meridian,  called  eastern  time,  included 
the  territory  between  the  Atlantic  coast  and  a  line  drawn 
through  Detroit,  Pittsburg,  Wheeling,  Parkersburg,  Hunt- 


ington, Bristol,  Tenn.,  Augusta,  G-a.,  and  Charleston,  these 
cities  being  the  termini  of  important  railroads.  Central 
time  is  governed  by  the  time  of  the  90th  meridian,  and  is 
used  by  the  section  west  of  Detroit,  etc.,  to  Bismarck,  North 
Platte,  Dodge  City,  etc.  The  next  section  which  takes  the 
time  of  the  105th  meridian,  called  mountain  time,  extends 
to  Helena,  Ogden,  and  the  western  boundary  of  Arizona. 
The  rest  of  the  country  to  the  Pacific  Ocean  takes  the  time 
of  the  120th  meridian,  called  Pacific  time. 


360  Chapter  Six. 

SOLAR  TIME. 
465.   Written  Exercises. 

1.  Find  the  difference  between  the  sun  time  of  London 
and  that  of  Chicago,  longitude  87°  35'  west  of  London. 

A  difference  of  15  degrees  of  longitude  makes  a  difference  of  1  hour ; 
of  16  minutes  of  longitude,  a  difference  of  1  minute ;  of  15  seconds  of 
longitude,  a  difference  of  1  second. 

If  1  degree  of  longitude  made  a  time  difference  of  1  hour,  the  differ- 
ence in  time  between  London  and  Chicago  would  be  87  hr.  35  min. ; 
as   it   takes  15    degrees   to   make   a 

difference  of  an  hour,  the  difference     15)87  hr.  35  min. 
of  time  between  London  and  Chicago  5  hr.  50  min.  20  sec. 

is  ^  of  87  hr.  35  min.   Dividing,  there- 
fore, 87  hr.  35  min.  by  15,  we  get  the  time  difference  as  5  hr.  50  min. 
20  sec. 

To  find  the  time  difference,  divide  the  longitude  difference 
etopressed  as  hours,  minutes,  and  seconds,  by  15. 

t.  When  it  is  midnight  at  London,  what  is  the  sun  time 
at  Chicago  ? 

Since  the  more  easterly  place  has  the  later  time,  it  is  5  hr.  50  min. 
20  sec.  before  midnight  at  Chicago.  12  hr.  (p.m.)  —  5  hr.  50  min.  20 
sec.  =  6  hr.  9  min.  40  sec.  p.m.    Ans. 

3.  Two  places  differ  in  longitude  37°  18V  What  is  their 
difference  in  solar  time  ? 

4.  Find  the  difference  in  longitude  between  two  places 
differing  in  solar  time  3  hr.  44  min. 

Multiply  3°  44'  by  15. 

To  find  the  longitude  difference,  multiply  by  IS  the  time 
difference  expressed  as  degrees,  minutes,  and  seconds. 

5.  Find  the  difference  in  sun  time  between  two  places  in 
longitude  74°  31'  and  93°  14'  west  of  Greenwich,  respec- 
tively. 


Solar  Time.  361 

6.  When  it  is  noon  at  a  place  11°  east  of  Greenwich,  it  is 
1 :  30  p.m.  at  another  place.  Find  the  longitude  of  the  latter 
place. 

Note.  —  Owing  to  the  general  use  of  standard  time  by  civilized 
countries,  problems  in  longitude  and  time  have  no  practical  value 
except  for  navigators.  The  following  problems  should  be  worked  only 
after  more  important  topics  have  been  completed. 

Note. — The  word  "time"  in  the  following  problems  means 
"mean  solar  time." 

7.  Given  the  longitude  of  A  as  95°  east,  and  that  of  B  as 
74°  east,  and  the  time  at  A  as  1:30  p.m.,  to  find  the  time  at  B. 

Since  the  latitude  of  B  has  no  bearing  upon  its  time,  both  places 
may  be  located  upon  the  same  line  running  east  and  west. 

Time  difference  =  ?  hours. 

Time=?    Time  1:30  p.m. 

B  A 

West 1 1 1 East 

0°  74°  95° 

Longitude  difference  =  21°. 

Locate  the  prime  meridian  (that  of  0°),  then  the  meridians  of  74° 
and  95°  east.  Mark  above  the  last  two  the  names  of  the  places,  B  and 
A.     Write  above  A  its  given  time,  1 :  30  p.m. 

To  find  the  time  at  B,  we  must  find  the  difference  of  time  between 
B  and  A.  The  difference  in  longitude  is  95°  -  74°  =  21°.  The  dif- 
ference in  time  is  21  hours  -f- 15. 

Note.  — Remember  that  the  more  easterly  of  the  two  places  has  the 
later  time. 

8.  A  is  situated  in  71°  west  longitude,  B  in  107°  west 
longitude.     What  time  is  it  at  B,  when  it  is  noon  at  A  ? 

Time  difference  =  ? 
Time  ?      12  m. 
B  A 

West 1 1 1 East 

107°         71°  0° 

Longitude  difference  =  ? 


362  Chapter  Six. 

9.    Find  the  longitude  of  B,  whose  time  is  8 :  10 :  30  a.m., 
when  it  is  7 :  15  a.m.  at  A,  whose  longitude  is  156°  48'  west. 

Time  difference  =  ? 
7:15  a.m.  8:10:30  a.m. 

A  B 

West 1 1 1  East 

156°  48'  Longitude  =  ?  0° 

Longitude  difference  =  ? 

Since  B  has  the  later  time,  its  location  is  east  of  A.  The  difference 
in  time,  being  nearly  an  hour,  shows  the  difference  in  longitude  to  be 
nearly  15°.  Find  the  exact  difference.  Is  it  to  be  added  to  166°  48' 
or  subtracted  from  it,  to  give  the  longitude  of  B  ? 

10.  When  it  is  2:40  a.m.  at  A,  in  57°  24'  west  longitude, 
it  is  10  a.m.  at  B.     Find  the  longitude  of  B. 

Time  difference  =  7\  hours. 

2:40  a.m.  10  a.m. 

A  B 

West 1 1 1  East 

57°  24'  0°  Longitude  =  ? 

Longitude  difference  =  15°  x7|  =  110°. 

If  we  go  110°  eastward  from  A,  we  shall  reach  the  prime  meridian 
after  going  how  many  degrees  and  minutes  ?  How  many  more  degrees 
and  minutes  must  we  travel  to  reach  B?  Is  B  in  east  or  in  west 
longitude  ? 

11.  When  it  is  noon  at  B,  what  is  the  time  at  A,  the 
former  being  in  longitude  44°  east,  and  the  latter  in  longi- 
tude 57°  west  ? 

Time  difference  =  ? 
Time  =  ?  12  m. 

A  B 

West 1 1 1 East 

67°  0°  44° 

Longitude  difference  =  101°.     Why  ? 


Solar  Time. 
Find  the  longitude  or  the  time : 


363 


Longitude  of  A. 

12.   63°  east 


Time  at  A. 
9  A.M. 


13.  57°  25'  east  ? 

14.  156°  48' west     3:15  p.m. 


15.  ? 

16.  2°  15'  west 

17.  27°  10' east 

18.  ? 

19.  74°  56' west 

20.  4'  30"  east 

21.  ? 


11:42  a.m. 
6:53  a.m. 
9 

4:10  p.m. 

3:50  a.m. 

8:47  a.m. 

10:30  p.m. 


Longitude  of  B. 

54°  east 
83°  20'  east 
? 

56°  25'  west 
67°:  48' east 
27°  10'  west 
18°  4'  east 
9 

90°  15'  west 
32°  30'  east 


Time  at  B. 
9 

1 :  45  p.m. 

4:10  p.m. 

1:27  p.m. 

9 

12  m. 
11:30  a.m. 

11  A.M. 
9 

6:48  p.m. 


REVIEW. 
466.   Oral  Problems. 

1.  At  what  per  cent  will  $12,  in  3  yr.  4  mo.,  amount  to 
$14? 

2.  What  will  be  the  cost  of  a  building  lot  100  feet  long 
and  50  feet  wide  at  50  ^  a  square  foot  ? 

3.  A  horse  was  sold  for  $90,  at  which  price  12^%  was 
gained.  What  per  cent  would  have  been  gained  by  selling 
him  for  $100? 

4.  What  is  the  premium  for  insuring  $6000  on  my 
house  at  1\%  ? 

5.  How  many  cubic  inches  in  a  ten-inch  cube  ? 

6.  Bought  2  chairs  at  $1.25,  one  wash-tub  for  $1.50, 
1  table  for  $3.00,  and  5  dozen  glasses  at  48^  a  dozen.  Gave 
a  ten-dollar  bill  in  payment.  How  much  change  did  I 
receive  ? 


364  Chapter  Six. 

7.  My  desk  is  1£  feet  long,  and  1  foot  wide.    How  many 
inches  around  it  ? 

8.  If  a  man  spends  50^  a  day  during  April,  May,  and 
June,  what  does  he  spend  in  the  three  months  ? 

9.  A  grocer  bought  15  barrels  of  flour  at  $5  a  barrel. 
At  what  price  must  he  sell  them  to  gain  $  36  ? 

10.  Seven-eighths  of  James's  vacation  will  be  equal  to 
seven-ninths  of  yours ;  yours  will  be  63  days.  How  many 
will  his  be  ? 

11.  A  man  sold  two  cows  for  $30  each.  On  one  he 
gained  25% ;  on  the  other  he  lost  25  %.  Did  he  gain  or  lose, 
and  how  much  ? 

12.  What  principal,  in  three  years  and  4  months,  at  6%, 
will  give  $40  interest? 

Note. — To  the  following  ten  problems  the  wrong  answers  are  very 
frequently  given. 

13.  Sold  a  horse  for  $  250,  losing  $  50.  What  is  the  loss 
per  cent  ? 

14.  If  3  boys  solve  3  problems  in  3  minutes,  how  long 
will  it  take  6  boys  to  solve  6  problems  ? 

15.  Two  boys  go  fishing;  one  brings  7  cakes  for  lunch, 
the  other  brings  5  cakes.  A  third  boy  joins  them  at  noon, 
and  pays  12^  for  his  share  of  the  meal.  How  should  the 
first  two  divide  the  money  received  ? 

16.  If  100  per  cent  is  gained  by  selling  an  article  for  $  1, 
how  much  would  be  gained  by  selling  it  for  $  2  ? 

17.  A  boy  had  a  slate  5  inches  by  7  inches.  He  buys 
one  twice  as  large.     Give  the  dimensions  of  the  new  slate. 

18.  A  man  wishes  to  put  up  on  the  front  of  his  lot  a 
fence  30  feet  long.  If  the  posts  are  6  feet  apart,  how  much 
will  they  cost  at  25^  each? 


Review.  365 

19.  One-half  the  money  taken  in  by  a  newsboy  is  profit. 
What  per  cent  does  he  make  ? 

20.  50  per  cent  of  a  number  multiplied  by  30  per  cent  of 
the  same  number  equals  60.     What  is  the  number  ? 

21.  Three-fourths  per  cent  of  a  number  is  90.  What  is 
the  number  ? 

22.  An  importer  receives  some  cases  of  goods  numbered 
consecutively.  How  many  cases  are  there,  if  the  number  of 
the  first  is  28,  and  of  the  last  75  ? 

467.   Written  Problems. 

1.  What  is  the  profit  on  9  boxes  of  oranges,  each  con- 
taining 20  dozen,  bought  at  $  1.10  per  hundred  and  sold  at 
the  rate  of  18  for  25^? 

2.  How  long  will  it  take  a  train  to  go  176  miles  at  the 

rate  of  3520  feet  per  minute  ? 

i 

3.  If  .0375  of  an  acre  of  land  is  worth  $  9,  what  is  -^ 

acre  worth  ? 

4.  At  £  1  Is.  Id.  per  barrel,  how  many  barrels  of  flour 
can  be  bought  for  £  161  17s.  6d.  ? 

5.  If  580  tiles,  each  6  inches  square,  will  cover  a  certain 
area,  how  many  tiles,  each  4  inches  long  and  3  inches  wide, 
will  be  needed  to  cover  the  same  area  ? 

6.  A  man  receives  $  1500  commission  on  his  yearly 
sales.  What  is  the  amount  of  his  sales  if  he  is  allowed  \ 
per  cent  commission  ? 

7.  At  what  rate  per  cent  will  $360  produce  $3.06  in- 
terest in  2  mo.  12  da.  ? 

8.  Find  the  square  root  of  25.00400016. 

9.  What  will  be  the  capacity,  in  gallons,  of  a  tank  9  ft 
long,  6  ft.  8  in.  wide,  and  6  ft.  5  in.  deep  ? 


366  Chapter  Six. 

10.  What  decimal  multiplied  by  312.5  will  give  the  sum 
of  I,  ^  t>  -09375,  and  2.46  ? 

11.  A  dealer  bought  a  lot  of  coal  $  4.95  per  ton.  What 
was  the  total  cost  if  he  gained  $  142.50  by  selling  it  at 
$  5.25  per  ton  ? 

12.  Find  the  value  of  ?i±iA_  1  0f  64. 

1|  x  3       ¥        T 

13.  The  front  wheel  of  a  wagon  measures  13  feet  in  cir- 
cumference. What  is  the  distance  travelled  in  miles,  rods, 
yards,  etc.,  when  the  wheel  has  made  527  revolutions  ? 

14.  Write  in  words  .349,  300.049,  $h%  300^^. 

15.  If  a  bar  of  silver  weighing  4  lb.  6  oz.  12  pwt.  is  worth 
£6  14s.  2d.,  what  is  the  value  (in  English  money)  of  a 
similar  bar  weighing  7  lb.  9  oz.  12  pwt.  ? 

16.  A  and  B  form  a  partnership.  A  furnishes  $5000; 
B,  $  10,000.  During  the  year  A  draws  $  1500  of  the  profits 
and  B  draws  $1000.  At  the  end  of  the  year  the  entire 
business  is  disposed  of  for  $  20,000.  What  amount  should 
each  receive  ? 

17.  What  per  cent  is  gained  on  an  article  bought  for  20 
per  cent  less  than  its  value  and  sold  for  20  per  cent  more 
than  its  value  ? 

18.  A  person  loans  $  750  to  M  and  $  1200  to  N  at  the 
same  rate.  From  the  latter  he  receives  half-yearly  $  9  more 
interest  than  from  the  former.  What  is  the  annual  rate  of 
interest  ? 

19.  A  4-months  note  for  $  375,  drawn  March  19,  was  dis- 
counted at  a  bank  June  4.     Find  the  proceeds.     Eate,  6%. 

20.  M  can  do  a  piece  of  work  in  4  days,  N  can  do  it  in  5 
days,  O  in  6  days.  How  long  will  it  take  the  three  to- 
gether to  do  the  work  ? 

1  day  +  (£  +  |  +  $).    Analyze. 


Stocks.  367 

STOCKS. 

468.  Some  undertakings,  such  as  the  construction  of  a 
railroad,  the  building  and  equipment  of  a  factory,  the  de- 
velopment of  a  mine,  and  the  like,  require  more  money  than 
any  individual  may  care  to  risk.  It  then  becomes  necessary 
to  secure  the  cooperation  of  a  number  of  persons. 

The  people  of  a  certain  town  desire  to  build  a  street  rail- 
road, the  construction  and  equipment  of  which  will  require 
$50,000.  The  projectors  organize  a  company.  If  it  is 
desired  to  interest  people  of  small  means,  the  required  capi- 
tal may  be  divided  into  shares  of  $10  each,  making  the  total 
number  of  shares  5000.  If  the  shares  are  fixed  at  $100 
each,  there  will  be  500  shares. 

To  every  purchaser  of  shares,  a  certificate  is  issued,  coun- 
tersigned by  the  officers  of  the  company,  setting  forth  the 
amount  of  capital,  the  total  number  of  shares,  and  the  num- 
ber issued  to  the  holder  of  the  certificate. 

At  certain  fixed  periods,  quarterly,  semi-annually,  or  an- 
nually, the  directors  of  the  company  determine  what  part  of 
the  profits  shall  be  distributed  to  the  stockholders,  the 
remainder  being  reserved  for  new  cars,  extension  of  the 
road,  etc.     The  profits  thus  distributed  are  called  dividends. 

469.  Written  Problems. 

1.  A  company  is  organized  with  a  capital  of  $50,000, 
divided  into  shares  of  $  100  each.  What  part  of  the  stock 
is  held  by  the  owner  of  10  shares  ? 

2.  If  dividends  of  $2000  are  distributed  at  the  end  of 
six  months,  how  much  should  the  holder  of  10  shares 
receive  ? 

3.  The  company  announces  the  dividend  as  a  certain  per 
cent  of  the  capital.  What  per  cent  dividend  is  declared  in 
this  case  ?     To  what  per  cent  per  year  is  it  equal  ? 


368  Chapter  Six. 

4.  Mr.  H.  has  $4500  in  the  savings  bank,  on  which  he 
receives  4  per  cent  interest.  He  gives  this  amount  for  30 
shares  of  the  stock.  What  price  does  he  pay  per  share? 
What  per  cent  of  the  par  value  ? 

5.  If  the  next  semi-annual  dividend  is  4%,  how  much 
more  income  does  Mr.  H.  receive  from  the  stock  than  he 
would  obtain  from  the  savings  bank  ? 

6.  What  per  cent  has  Mr.  H.  received  for  six  months  on 
his  investment  of  $4500  ? 

7.  If  Mr.  H.  sells  the  30  shares  at  $164.50  per  share, 
how  much  more  does  he  receive  for  it  than  it  cost  him  ? 

470.  Stocks  are  generally  bought  and  sold  by  brokers, 
who  charge,  as  a  rule,  J%  of  the  par  value  for  buying  or  for 
selling.  The  prices  of  the  stocks  as  given  in  the  news- 
papers are  generally  a  percentage  of  the  par  value.  Thus, 
the  New  York  quotation  of  Pennsylvania  E..R.  on  March 
4,  1903,  is  151-J-.  This  means  that  the  shares  of  the  Penn- 
sylvania R.R.  sold  for  $50x1.51^,  or  $75.75,  the  par 
value  being  $50.  The  Philadelphia  papers  of  the  same 
date,  however,  quote  the  stock  at  75f ,  it  being  the  practice 
in  that  city  to  give  the  price  per  share. 

471.  "Written  Exercises. 

1.  Find  the  cost  of  240  shares  Anaconda  Copper  Mining 
Co.,  par  value  $25,  at  134|,  brokerage  \%. 

Cost  =  $25  x  240  x  (1.34$  +  .00J). 

To  find  the  cost,  multiply  the  face  value  of  the  given  number 
of  shares  by  the  rate  plus  the  brokerage. 

2.  How  much  brokerage  is  paid  by  the  buyer  of  275 
shares  bank  stock,  par  value  $100,  brokerage  £%? 

|  %  of  $100  x276. 


Stocks.  369 

3.  Paid  $11,445  for  120  shares  Cleveland,  Cincinnati, 
Chicago,  &  St.  Louis,  par  value  $100,  brokerage  £%.  What 
was  the  value  of  the  stock  per  share  ? 

The  brokerage  on  120  shares,  par  value  $  100,  is  £%  of  $  12,000,  or 
$15.  The  cost  of  the  stock  is,  therefore,  $11,430.  Dividing  by  the 
number  of  shares  gives  the  value  per  share. 

($  11,445  -  }  %  of  [$  100  x  120])  -s- 120. 

4.  Bought  150  shares  Evansville  and  Terre  Haute  at 
69f,  brokerage  \%,  paying  for  it  $5212.50.  What  is  the 
par  value  per  share  ? 

The  cost  of  each  share  is  $  5212.50  -*- 150.  Divide  this  cost  by  the 
rate,  including  the  brokerage,  .69|  +  .00$. 

($ 5212.50  -r- 150)  +  (.69f  +  .00$). 

5.  A  broker  sells  for  a  customer  200  shares  stock,  par 
value  $25,  at  102 J.  If  he  retains  -$%  brokerage,  how  much 
does  he  pay  over  to  the  former  owner  of  the  stock  ? 

6.  A  man  buys  60  shares  bank  stock,  par  value  $  100,  at 
450,  no  brokerage.  If  the  annual  dividend  is  18%,  what  is 
his  income  therefrom  ?  What  per  cent  does  he  receive  on 
his  investment  ? 

Note.  —  Dividends  are  based  upon  the  par  value. 

7.  A  manufacturing  corporation  makes  $20,000  a  year 
over  all  expenses.  The  stock  consists  of  4000  shares,  par 
value  $  50.     What  rate  of  dividend  can  be  declared  ? 

What  per  cent  on  his  investment  does  a  man  receive  who 
has  bought  his  stock  at  175,  no  brokerage  ? 

8.  A  capitalist  bought  360  shares  stock,  par  value  $  25, 
at  168£.  He  paid  therefor,  including  brokerage,  $  15,176.25. 
What  was  the  rate  of  brokerage  ? 

9.  A  broker  sold  250  shares,  par  value  $100,  at  107f. 
He  deducted  brokerage  and  paid  over  the  proceeds,  amount- 
ing to  $  26,875.  Find  the  amount  of  the  brokerage  and  the 
rate  per  cent. 


370  Chapter  Six. 

10.  A  woman  invests  $  35,050  in  stock  at  175,  brokerage 
J%.  If  the  annual  dividends  are  7^-%,  what  is  her  income 
from  the  investment  ? 

11.  Which  investment  will  pay  better,  one  in  a  gas  com- 
pany paying  6%  dividends  annually,  their  stock  selling  at 
150,  the  other  in  a  bank  paying  7%  dividends  annually, 
stock  selling  at  175  ? 

12.  What  annual  dividend  should  be  declared  on  railroad 
stock  bought  at  125,  so  that  the  buyer  will  receive  4%  per 
annum  on  his  investment  ?     What  semi-annual  dividend  ? 

13.  What  will  be  the  cost  of  17  shares  of  canal  stock,  par 
value  $  50,  at  93  J,  and  143  shares  gas  stock,  par  value  $  10, 
atl02f? 

Note.  —  An  examination  of  the  prices  of  stocks  as  given  in  the 
newspapers  will  show  that  the  rate  of  dividends  constitutes  but  one 
consideration  influencing  buyers.  The  following  prices  were  offered 
March  4,  1903,  f cr  stocks  of  four  banks,  respectively,  each  of  which 
paid  6  per  cent  dividends  annually  ;  185,  245,  390,  685.  Purchasers  of 
shares  of  the  last  three  banks  evidently  hoped  for  larger  dividends  in 
the  immediate  future. 

.  The  values  of  bonds  depend  in  the  first  instance  upon  the  character 
of  the  corporation  issuing  them,  then  upon  the  rate  of  interest  and  the 
length  of  time  before  redemption.  United  States  bonds  bring  the 
highest  prices,  as  buyers  have  no  fear  of  the  failure  of  the  government 
to  keep  its  promises.  The  following  are  the  prices  obtained  at  the  last 
sales  reported  to  March,  1903 : 


Date  of 

Rate. 

Redemption. 

Price  Paid. 

Last  Sale. 

U.  S.  2's 

1930 

108f 

Nov.  14,  1902 

U.  S.  4's 

1907 

no* 

Feb.    4,  1903 

U.  S.  4's 

1925 

136 

Feb.  26,  1903 

U.  S.  5's 

1904 

103 

Feb.  23,  1903 

The  5  per  cent  bonds,  although  bearing  the  highest  rate  of  interest, 
bring  only  103,  as  they  will  be  redeemed  at  par  a  little  more  than  a 
year  after  they  are  bought.  The  purchaser,  who  paid  $  103  for  a  bond, 
will  receive  for  it  in  1904  only  $100,  with  about  $5  interest,  his  net 
profit  for  the  year  being  $2  on  an  investment  of  $  103. 


Bonds.  371 

BONDS. 

472.  A  bond  is  a  form  of  interest-bearing  note  issued  by  a 
corporation. 

A  coupon  bond  is  one  containing  certificates  of  interest 
which  are  cut  off  and  presented  for  payment  as  interest 
becomes  due.  A  10  years'  U.  S.  coupon  bond  has  40  cou- 
pons, one  for  each  quarter-year's  interest.  Upon  each  is 
engraved  the  date  when  due,  and  the  sum  payable,  which 
is  $  10  in  the  case  of  a  $  1000  f our-per-cent  bond. 

A  registered  bond  contains  no  coupons,  a  check  for  the 
interest  being  mailed  to  the  owner,  whose  name  is  registered 
on  the  books  of  the  corporation. 

473.  Written  Exercises. 

1.  A  railroad  company  needing  more  money  to  extend  its 
road,  issues  bonds  bearing  interest  at  4%.  If  these  bonds 
are  sold  at  95,  what  rate  of  interest  on  the  money  invested 
does  the  owner  of  a  bond  receive  ? 

For  each  $95  invested  the  owner  receives  $4  interest.  The  rate  is 
4  -4-  .95. 

To  find  the  rate  on  the  investment,  divide  the  rate  of  interest 
by  the  rate  paid  for  the  bond,  including  brokerage,  if  any. 

2.  Find  the  cost  of  20  one  thousand  dollar  bonds  at  120£, 
brokerage  §-%. 

3.  If  the  foregoing  bonds  bear  interest  at  the  rate  of  6%, 
what  is  the  annual  income  ?  What  rate  per  cent  annually 
is  received  on  the  sum  invested  ? 

4.  A  man  desires  to  secure  an  annual  income  of  $  650  for 
his  daughter.  What  is  the  face  value  of  5%  bonds  necessary 
to  produce  this  income?  What  will  be  the  cost  of  5% 
bonds  of  Denver  &  Rio  Grande  at  107,  brokerage  l%? 


372  Chapter  Six. 

5.  A  person  desirous  of  obtaining  a  semi-annual  income 
of  $900  is  offered  Central  Pacific  4's  at  99  J,  Chicago  &  Alton 
3's  at  83  J,  or  Western  Union  4£'s  at  104 J,  no  brokerage  in 
any  case.  Find  the  difference  between  the  smallest  and  the 
largest  outlay  necessary  to  secure  the  desired  income  from 
these  bonds. 

Note.  — 4's  means  bonds  paying  4  per  cent  interest  per  year. 

6.  How  much  money  must  be  invested  in  the  U.  S.  2's  to 
yield  a  quarterly  income  of  $225,  bonds  selling  at  108|, 
brokerage  \°/0  ? 

7.  An  owner  of  6  per  cent  bonds  sells  them  at  the  market 
quotation  of  118,  and  invests  the  proceeds  in  4£  per  cent 
bonds.  The  latter  investment  yields  him  the  same  income  as 
the  former.  What  did  he  pay  per  hundred  for  the  4^-  per 
cent  bonds,  no  brokerage  ? 

8.  A,  having  a  farm  of  109  acres,  which  rents  for  $  681.25 
above  taxes,  etc.,  sells  the  same  for  $200  per  acre,  and 
invests  the  proceeds  in  U.  S.  2's  @  108£%,  brokerage  |%. 
Will  his  yearly  income  be  increased  or  diminished,  and  how 
much? 

9.  What  is  the  difference  in  the  rate  of  income  obtained 
from  an  investment  in  U.  S.  2's  at  109|,  and  one  in  U.  S.  4's 
at  137f,  brokerage  \%  in  each  case? 

Note.  —  In  calculating  the  rate  of  interest  in  the  foregoing  exam- 
ples, the  time  of  the  redemption  of  the  bonds  is  omitted  from  consid- 
eration. In  the  following  example,  however,  the  term  of  the  bond  is 
made  an  element  in  the  computation.  The  holder  of  it  has  received 
$30  in  interest,  and  he  is  paid  $100  for  the  bond.  Ignoring  the  mat- 
ter of  compound  interest,  the  question  becomes:  At  what  rate  will 
$  104  amount  in  6  years  to  $  130  ?  * 

10.  Mr.  Tower  pays  $104  for  a  $100  five  per  cent  bond. 
At  the  end  of  six  years  the  bond  is  redeemed  at  par.  What 
rate  of  interest  does  he  receive  on  his  investment  of  $  104  ? 


Exchange.  373 

DOMESTIC  EXCHANGE. 

474.  Arthur  S.  Somers,  of  Memphis,  Tenn.,  wishes  to  pay- 
John  R.  Thompson,  of  The  City  of  New  York,  $  3475.86.  If 
Mr.  Somers  sends  a  check,  drawn  on  his  Memphis  bank, 
Mr.  Thompson  will  be  charged  a  certain  sum  by  his  New 
York  bank  for  collecting  the  amount  of  the  check,  and  he 
will  thus  receive  somewhat  less  than  the  sum  due  him.  Mr. 
Somers,  therefore,  buys  from  J.  E.  Washington,  a  Memphis 
banker,  who  has  funds  in  a  New  York  bank,  the  following 

SIGHT   DRAFT. 

f  3475 J£6^.  Memphis,  Tenn.,  Aug.  9,  1904. 

At  sight,  pay  to  the  order  of  John  R.  Thompson  Three 
Thousand  Four  Hundred  Seventy-five  -fa  Dollars,  value 
received,  and  charge  to  the  account  of 

To  Chemical  Bank,  JOSEPH   E.  WASHINGTON. 

The  City  of  New  York. 

Mr.  Somers  is  charged  for  this  draft  a  premium  of  $  1.50 

per  $  1000;  that  is,  he  pays  Mr.  Washington  $1001.50  for 

each  $  1000.     The  cost  of  the  draft  is,  therefore,  $  3475.86 

X  1.0015,  or  $  3481.07. 

475.  Exchange  is  at  a  premium  when  the  cost  of  a  sight 
draft  is  greater  than  its  face;  it  is  at  a  discount  when  the 
cost  of  a  sight  draft  is  less  than  its  face. 

476.  Mr.  Thompson  could  collect  the  sum  due  him  by 
making  a  draft  on  Mr.  Somers  as  follows : 

TIME    DRAFT. 
$  3475_8T6r.  New  York,  Aug.  9,  1904. 

At  three  days'  sight  pay  to  the  order  of  The  National 
Bank  of  Commerce  Three  Thousand  Four  Hundred  Seventy- 
five  and  ■££$  dollars,  value  received,  and  charge  to  the 
account  of  JoHN  R#  Thompson. 

To  Arthur  S.  Somers, 
Memphis,  Tenn. 


374  Chapter  Six. 

Mr.  Thompson  deposits  the  draft  in  the  National  Bank  of 
Commerce  for  collection.  This  bank  forwards  it  to  a  Mem- 
phis bank.  The  latter  notifies  Mr.  Somers.  If  he  wishes 
to  pay  the  draft  at  the  expiration  of  three  days,  he  writes 
across  the  face  in  red  ink,  "Accepted,"  with  the  date, 
"  Aug.  11,  1904/'  and  adds  his  signature.  Aug.  14  he  pays 
the  money  to  the  Memphis  bank,  which  notifies  the  Bank 
of  Commerce,  and  the  sum  is  placed  to  the  credit  of  Mr. 
Thompson,  less  the  cost  of  collection. 

477.  A  sight  draft  is  payable  upon  presentation,  except 
in  those  states  allowing  "  days  of  grace."  A  time  draft  is 
one  payable  a  specified  number  of  days  after  acceptance. 
In  some  states  three  additional  "  days  of  grace "  are 
allowed. 

478.  Written  Exercises. 

1.  Find  the  cost  of  a  Boston  draft  on  New  York  for 
$  1875,  at  12  ^  discount  per  $  1000. 

Face  $  1875. 

Discount  $  1875  x  .00012  .225 

$1874.775 
Ans.  $  1874.78. 

To  find  the  cost  of  a  sight  draft,  add  the  premium  to  the 
face,  or  subtract  the  discount  from  the  face. 

2.  What  will  a  St.  Louis  merchant  pay  for  a  draft  on 
New  York  for  $  2460.53,  at  50  ^  premium  per  $  1000  ? 

3.  At  \°lo  premium,  find  the  cost  of  a  sight  draft  for 
$1843.60. 

4.  At  75^  discount  per  $1000,  how  much  will  cost  a 
sight  draft  on  Milwaukee  for  $946.75? 

5.  Paid  $  632.18  for  a  sight  draft  on  Milwaukee.  What 
was  the  face  of  the  draft,  the  discount  being  ■&%? 


Exchange.  375 


BILLS  OF  EXCHANGE. 

479.  Bills  of  exchange  are  either  domestic  or  foreign.  A 
domestic  bill  of  exchange  is  called  a  draft,  the  term  bill  of 
exchange  being  generally  applied  only  to  foreign  bills. 

480.  Fred  Johnston  owes  John  Ahern  &  Co.,  of  London, 
£  180  17s.  6d.  He  buys  from  John  Cottier  &  Brother  a 
bill  of  exchange  drawn  on  their  London  correspondent. 
The  bill  is  drawn  in  duplicate,  one  being  sent  by  Mr. 
Johnston  to  John  Ahern  &  Co.,  and  the  other  being  retained 
by  the  former  to  send  in  case  of  the  loss  of  the  first.  When 
either  is  paid  the  other  becomes  of  no  value. 

The  following  is  the  form  of  the  first  of  a  set  of  exchange. 

Exchange  for  £  180  17s.  6d.  New  York,  Dec.  14,  1903. 

Sixty  days  after  sight  of  this  First  of  Exchange  (Second 
unpaid),  pay  to  the  order  of  John  Ahern  &  Co.,  One  Hun- 
dred Eighty  Pounds  Sterling,  Seventeen  Shillings  Six 
Pence,  value  received,  and  charge  the  same  to  account  of 

To  James  Lennon  &  Co.,         John  Cottier  &  Brother. 
London. 
No.  39. 

Upon  receipt  of  this  bill,  John  Ahern  &  Co.  present  it 
for  acceptance.  They  receive  the  money  sixty  days  there- 
after. 

481.  Written  Exercises. 

1.    Find  the  cost  of  the  above  bill  at  $4.87  per  pound. 

£200  =  $974.00 
20  =       97.40 
£  180  =  $ 
10s.  =      2.435  ££ 
5s.  = 
2s.  6d.  = 


376  Chapter  Six. 

2.  What  is  the  cost  of  a  cable  transfer  of  £251  lis.  9&, 
at  $4.88£  per  pound? 

£250  =  ^1221.25    \  of  £1000 

1  = 

10s.  = 

Is.  = 

6d.= 

3d.  = 

The  newspapers  give  quotations  of  foreign  exchange  for  sight  and 
60-day  bills,  also  for  cable  transfers. 

482.  The  New  York  quotations  for  French  exchange  give  the 
number  of  francs  for  $1. 

Paris  cable  transfers        5. 16 \  @  5.15f. 

Paris  bankers' 60  days     5.18f@5.18|. 

Paris  bankers'  sight         5.16|  @  5. 16  J. 

The  quotations  for  German  exchange  give  the  value  in  U.  S.  money 
of  4  Reichmarks  (or  marks). 

Reichmarks  (4)  60  days    95J  @  95J. 

Reichmarks  (4)  sight        95|  @  95$. 

3.  Find  the  cost  of  a  sight  bill  on  Paris  for  1000  francs, 
at  5.16J  francs  for  $  1. 

4.  Find  the  cost  of  a  60-day  bill  of  exchange  on  Berlin 
for  1874.35  marks,  at  95J  fi  for  4  marks. 

5.  What  will  be  the  face  in  marks  of  a  sight  bill  of  ex- 
change on  Berlin  that  can  be  bought  for  $  1000,  at  95 J  ^  for 
4  marks  ? 

6.  A  New  York  merchant  pays  $  1637.50  for  a  60-day 
bill  on  Paris.  What  is  the  face  of  the  bill,  the  rate  of  ex- 
change being  5.18J  francs  for  $  1  ? 


Exchange.  377 

7.  At  $4.88  per  pound,  what  will  be  the  face  of  the 
sight  bill  on  London  that  can  be  bought  for  $  1500  ? 

18750  *  £307  7s.,  etc. 

g||  =  18750  61)18750 

i&  61  _450 

to  £23  remainder 

20_ 
460s.,  new  dividend 

8.  Bought  goods  in  London  amounting  to  £  437  5s.  10c?. 
less  4%.  How  much  do  I  pay  in  Boston  for  a  sight  bill  of 
exchange  at  $  4.88^  to  settle  the  account  ? 

9.  What  will  be  the  cost  in  Chicago  for  a  60-day  bill  on 
Paris  that  will  pay  for  the  following  articles  ?  Rate,  1  franc 
-10}  A 

18  pieces  silk,  44  meters  each,  at  25  francs  per  meter, 
less  1\% 

3  pieces  of  cloth,  50  meters  each,  at  20  francs  per  meter, 
less  5%. 

Packing  charges,  60.50  francs. 

10.  I  wish  to  send  a  sight  bill  of  exchange  on  Berlin  in 
payment  of  the  following  invoice : 

4  cases  musical  instruments,  amounting  to  3598.60  marks, 
less  10,  5,  and  2\%. 

Freight  to  Hamburg,  165  kilos,  at  4.80  marks  per  kilo. 
At  95J  ^  for  4  marks,  what  will  be  the  cost  of  the  bill  of 
exchange  ? 

11.  If  the  rate  of  exchange  is  50^  discount  per  $1000, 
what  is  the  face  of  the  sight  draft  on  Boston,  that  can  be 
bought  in  New  York  for  $  1000  ? 

Note.  —  $  999.50  in  New  York  will  buy  a  sight  draft  on  Boston  for 
$1000. 

12.  When  the  premium  is  $1.25  per  $1000,  Mr.  Brown 
pays  $  1634.04  for  a  draft  on  Louisville.  What  is  the  face 
of  the  draft? 


378  Chapter  Six. 

COMPOUND  INTEREST. 

483.  Compound  Interest  is  interest  on  the  principal  and  on 
the  unpaid  interest,  which  is  added  to  the  principal  at  regu- 
lar intervals.  The  interest  may  be  compounded  annually, 
semi-annually,  or  quarterly,  according  to  agreement. 

Compound  interest  is  allowed  by  savings  banks.  It  is 
not  collectible  on  notes,  mortgages,  or  the  like. 

484.  "Written  Exercises. 

1.  Find  the  amount  of  $375,  for  1  year,  at  6%.  Con- 
sidering this  as  a  new  principal,  find  the  amount  for  a  year, 
same  rate.  Find  the  amount  of  this  last  principal  for  3 
months. 

2.  What  is  the  amount  of  $  375,  for  2  yr.  3  mo.,  at  6%, 
compound  interest  ? 

3.  What  is  the  amount  of  $  375,  for  2  yr.  3  mo.,  at  6%, 
the  interest  compounded  semi-annually  ? 

Principal,  $375. 

3%      11.25       6  months'  interest 

386.25       Amount  6  months. 

3%      11.5875  6  months' interest. 

Amount  1  year, 
3%  6  months'  interest. 

Amount  1£  years, 
etc.,  etc.,  etc. 

4.  Find  the  compound  interest  on  $  375,  for  2  yr.  3  mo., 
at  6  per  cent,  compounded  semi-annually. 

Note.  —  To  find  the  compound  interest,  deduct  $  375  from  the 
amount  for  2  yr.  3  mo. 

5.  What  is  the  amount  of  $  100,  at  compound  interest, 
for  3  years,  interest  at  6%,  compounded  annually  ? 


Annual  Interest.  379 

ANNUAL  INTEREST. 

When  the  maker  of  a  note  fails  to  keep  his  contract  to  pay  interest 
annually,  the  laws  of  some  states,  including  Michigan,  permit  the  col- 
lection of  simple  interest  on  the  deferred  payments  of  interest. 

485.   Written  Problems. 

1.  Find  the  amount  due  June  1,  1908,  on  the  following 
note,  no  payments  of  principal   or  interest   having   been 

Detroit,  Mich.,  June  1,  1904. 

Tour  years  after  date,  without  days  of  grace,  I  promise  to 
pay  to  the  order  of  Daniel  W.  Lawler,  Six  Hundred  Dollars, 
value  received,  with  annual  interest  at  six  per  cent. 

$600^.  George  Oxj^ard. 

Principal,  $600.00 

Interest,  4  years,  at  6  %,  144.00 

3  years'  interest,  at  6  %,  on  the  1st  year's  interest,  $  36,        6.48 
2  years'  interest,  at  6  %,  on  the  2d  year's  interest,  $  36, 

1  year's  interest,  at  6  %,  on  the  3d  year's  interest,  $  36, 

Amount  due  June  1,  1910,  $ 

Find  the  interest  on  the  principal  for  the  entire  time,  and  on 
each  annual  interest  for  the  time  it  remained  unpaid.  The 
sum  of  the  principal  and  all  the  interest  is  the  amount  due. 

2.  Find  the  amount  due,  at  5%,  for  5  years,  on  a  note 
for  $  1200,  annual  interest  being  unpaid. 

3.  The  maker  of  a  note  for  $  900,  with  annual  interest 
at  7%,  makes  the  first  and  the  second  interest  payments 
when  due.  How  much  will  he  owe  at  settlement,  6  years 
after  the  date  of  the  note  ? 

4.  Find  the  difference  between  the  amount  due  at  6%  for 
3  years  on  a  note  for  $  300,  annual  interest  unpaid,  and  the 
amount  of  the  same  sum  placed  at  compound  interest  for 
the  same  time  at  the  same  rate. 


380  Chapter  Six. 

5.  What  is  the  amount  of  a  note  for  $  720,  at  4  years,  at 
4|%,  annual  interest  unpaid  after  the  first  year? 

6.  Find  the  amount  due  March  1,  1906,  on  a  note  for 
$500,  dated  March  1,  1900,  with  interest  at  6%,  annual 
interest  unpaid  after  the  third  year. 

METRIC  SYSTEM. 

486.  The  metric  system,  which  is  used  in  nearly  all  the 
countries  of  continental  Europe,  is  based  upon  the  meter. 
The  length  of  the  meter  is  one  ten-millionth  part  of  the 
length  of  the  meridian  from  the  equator  to  the  poles  — 
about  39.37  inches. 

The  subdivisions  of  the  meter  are  denoted  by  the  Latin 
prefixes  milli  (tuW)»  centi  (y-J-^),  deci  (y1^).  For  the  multi- 
ples, the  Greek  prefixes  deka  (10),  hecto  (100),  kilo  (1000), 
and  myria  (10,000)  are  used. 

487.  It  will  be  noticed,  in  the  table  below,  that  small 
letters  are  used  for  the  abbreviations  of  the  Latin  prefixes 
of  the  subdivisions,  and  capital  letters  for  the  Greek  pre- 
fixes of  the  multiples.     The  following  is  the  table  of 

488.  Measures  of  Length. 

10  millimeters  (mm)        1  centimeter  (cm) 

10  centimeters  1  decimeter  (dm) 

10  decimeters  1  meter  (m) 

10  meters  1  dekameter  (Dm) 

10  dekameters  1  hectometer  (Hm) 

10  hectometers  1  kilometer  (Km) 

10  kilometers  *  1  myriameter  (Mm) 

The  units  of  this  table  in  common  use  are  the  centimeter,  the  meter, 
and  the  kilometer. 

Long  distances  are  expressed  in  kilometers.  The  thickness  of  wire 
is  given  in  millimeters. 


Metric  System.  381 

489.   Written  Problems. 

1.  What  will  be  the  cost  in  francs  of  380  m  75  of  dress 
goods  at  2  f  60  per  meter  ? 

380  m  75  is  read  380  meters  75  centimeters.  It  is  also  written 
380.75  m,  but  the  first  method  is  the  more  common  one  in  Europe. 
2  f  60  is  read  2  francs  60  centimes.  A  period  (.)  is  not  used  after  the 
abbreviations  of  meter,  liter,  franc,  etc. 

2.  How  many  square  meters  in  a  piece  of  carpet  26  m  50 
long,  85  cm  wide  ? 

3.  How  many  square  meters  in  a  circle  whose  diameter 
is  15  meters  ? 

4.  An  are  is  a  surface  10  meters  long,  10  meters  wide. 
How  many  ares  in  a  field  135  meters  long,  69  meters  wide  ? 

5.  Find  the  area  in  ares  of  a  right-angled  triangle  whose 
base  is  245  meters,  hypotenuse  875  meters. 

6.  A  stere  is  a  cubic  meter.  What  will  be  the  cost,  at 
8  f  50  per  stere,  of  a  pile  of  wood  10  meters  long,  1  meter 
wide,  3.25  meters  high  ? 

7.  A  cube  one  decimeter  each  way  contains  a  liter  (1), 
which  is  the  principal  unit  of  dry  and  liquid  measure. 

How  many  liters'  capacity  has  a  tank  10  m  50  long,  8  m 
wide,  6  m  50  high  ? 

Change  each  dimension  to  decimeters. 

8.  How  many  bottles,  each  containing  0 1  75,  can  be  filled 
from  a  hogshead  containing  222  1  ? 

9.  How  much  will  be  received  for  36  bags  of  beans,  each 
containing  68  liters,  at  1  mark  25  per  dekaliter  ? 

10.  A  liter  of  water  weighs  a  kilogram  (1000  grams). 
How  many  kilos  of  oil  would  a  tank  contain,  its  dimensions 
being  5  meters  by  4  meters  by  3  meters,  the  weight  of  the 
oil  being  92%  of  the  weight  of  water? 

11.  Assuming  the  length  of  the  meter  as  39.37  inches, 
what  is  the  length  of  the  kilometer  in  yards  ? 


382  Chapter  Six. 

.  490.   Measures  of  Surface. 

100  sq.  mm  =  1  sq.  cm 
100  sq.  cm  =  1  sq.  dm 
100  sq.  dm  =  1  sq.  m  =  1.196  sq.  yd. 

491.  The  square  meter  is  the  principal  unit  of  surfaces, 
such  as  walls,  ceilings,  floors,  etc. 

100  centiares  (ca)  =  1  are  (a)  =  119.6  sq.  yd. 
100  ares  =  1  hectare  (Ha)  =  2.47  acres. 

The  are  is  the  principal  unit  of  surface  of  small  plots  of 
land.  The  area  of  a  farm  is  expressed  in  hectares,  of  a 
country  in  square  kilometers. 

492.  Measures  of  Volume. 

1000  cu.  mm  =  1  cu.  cm 
1000  cu.  cm  =  1  cu.  dm 
1000  cu.  dm  =  1  cu.  m  =  35.316  cu.  ft. 

The  principal  unit  is  the  cubic  meter. 

493.  The  stere  (cubic  meter)  is  used  for  measuring  wood. 
10  decisteres  (dst)  =  1  stere  (st)  =  35.316  cu.  ft. 

10  steres  =  1  dekastere  (Dst). 

The  stere  is  the  only  unit  used. 

494.  Dry  and  Liquid  Measures. 

10  milliliters  =  1  centiliter. 

10  centiliters  =  1  deciliter.  Dry.  Liquid. 

10  deciliters    =  1  liter  (1)    =    .908  qt.  =    1.057  qt. 

10  liters  =  1  dekaliter  =  1.135  pk.  =   2.642  gal. 

10  dekaliters  =  1  hectoliter  =  2.837  bu.  =  26.417  gal. 

10  hectoliters  =  1  kiloliter. 

10  kiloliters     =  1  myrialiter. 

The  liter  and  the  hectoliter  are  the  principal  units. 


Review.  383 

495.  Table  of  Weight. 

10  milligrams  (mg)  =  1  centigram. 

10  centigrams  =  1  decigram. 

10  decigrams  =  1  gram  (gr). 

10  grams  =  1  dekagram. 

10  dekagrams  =  1  hectogram. 

10  hectograms  =  1  kilogram  (kilo)  =  2.2046  lb. 

10  kilograms  (Kg.)  =  1  myriagram. 

10  myriagrams  =  1  quintal. 

10  quintals  =  1  tonneau  (ton). 

The  kilo  is  the  ordinary  unit.  Heavy  articles  are  sold  by  the 
tonneau. 

496.  Written  Exercises. 

1.  The  Eiffel  tower  is  300  meters  high.  What  is  its 
height  in  feet? 

2.  The  Danube  is  2600  kilometers  long.  Find  its  length 
in  miles. 

3.  A  bottle  filled  with  water  weighs  1.170  kilos;  the 
weight  of  the  bottle  is  420  grams.  What  is  the  capacity 
of  the  bottle  in  liters? 

4.  Find  the  weight  in  kilos  of  15  liters  of  olive  oil,  which 
weighs  .915  time  as  much  as  water. 

5.  A  rectangular  field  123  meters  long,  and  85.5  meters 
wide,  yielded  13.25  hectoliters  of  wheat  per  hectare.  The 
wheat  weighed  84.350  kilos  per  hectoliter  and  sold  for  23.50 
francs  per  100  kilos.     What  sum  did  the  crop  bring  ? 

6.  What  will  be  the  cost  in  francs  of  papering  a  room 
5  m  42  long,  4  m  18  wide,  and  3  m  10  high,  at  1  f  20  per 
square  meter  ? 

7.  Calculate  the  expense  of  building  a  wall  14  m  50  long, 
7  m  80  high,  0  m  22  thick,  of  bricks  0  m  22  long,  0  m  11 
wide,  0  m  06  thick,  the  bricks  costing  58  francs  per  thou- 
sand and  the  labor,  etc.,  32  f  80  per  cubic  meter. 


384  Chapter  Six. 

8.  Find  the  profit  on  a  pile  of  wood  20  meters  long,  4 
meters  wide,  8  meters  high,  bought  at  12  francs  per  stere, 
and  sold  at  4  francs  per  100  kilos,  the  weight  of  the  wood 
being  .42  times  the  weight  of  water. 

9.  A  liter  of  wheat  weighs  760  grams.  When  ground  it 
produces  89  per  cent  flour  and  11  per  cent  bran.  Find  the 
weight  of  the  flour  that  can  be  made  from  the  wheat  con- 
tained in  a  bin  2  m  60  long,  2  m  40  wide,  and  1  m  50  deep. 
Find  the  value  of  the  wheat  at  4  f  85  per  double  dekaliter. 

10.  If  sea  water  contains  -£$  of  its  weight  of  salt,  how 
many  hectoliters  of  sea  water  should  be  evaporated  to  obtain 
100  kilos  of  salt,  a  liter  of  sea  water  weighing  1.026  kilos  ? 

REVIEW. 
497.   Oral  Problems. 

1.  If  I  yard  costs  $4.50,  what  will  £•  yard  cost? 

2.  If  3  men  can  do  a  piece  of  work  in  4  days,  how  long 
will  it  take  24  men  to  do  it  ? 

3.  What  principal  at  interest  for  5  years,  at  6  per  cent, 
will  produce  $  12,  simple  interest  ? 

4.  A  stack  of  hay  will  keep  a  cow  20  weeks,  or  a  horse 
15  weeks.     How  long  will  it  keep  them  both  ? 

Note.  —  What  part  will  each  eat  in  a  week  ?  What  part  will  both 
eat  in  a  week  ? 

5.  How  many  days  from  May  16  to  July  5  ? 

6.  Sold  a  cow  for  $24,  losing  thereby  40%  of  the  cost 
price.  Had  I  sold  her  for  33^%  advance  on  the  cost,  what 
should  I  have  received  for  her  ? 

7.  What  will  460  pounds  of  tea  cost  at  $  .48  per  pound  ? 

8.  If  12  ounces  of  bread  are  destroyed  in  making  a  gill  of 
whiskey,  how  much  will  be  destroyed  in  making  a  gallon  ? 

4  gills  =  1  pint 


Review.  385 

9.   If  the  weight  of  air  is  15  pounds  on  the  square  inch, 
what  is  it  on  the  square  foot  ? 

10.  Seven  is  three-fifths  of  what  number  ? 

11.  What  is  the  value  of  960  pounds  of  wheat  at  $1.05 
per  bushel  of  60  pounds  ? 

12.  At  what  rate  per  cent  will  $400  make  $37.50,  simple 
interest,  in  1  yr.  3  mo.  ? 

13.  What  is  the  brokerage  on  $10,400,  at  If  %  ? 

14.  What  will  3280  feet  of  lumber  cost  @$25  per 
thousand? 

15.  A  and  B  are  partners;  A  puts  in  -^  of  the  stock,  and 
B  the  remainder ;  B's  gain  is  $  1400.     Find  A's  gain. 

16.  What  is  the  difference  in  the  longitude  of  two  places 
whose  difference  in  sun  time  is  two  hours  and  three  minutes? 

17.  A  room  is  f  as  wide  as  it  is  long.  Its  length  is  20 
feet.     How  many  square  feet  are  there  in  the  floor  ? 

18.  If  5  yards  of  cloth  cost  90^,  what  will  f  of  a  yard 
cost? 

19.  An  agent  insured  a  house  for  me  at  a  commission  of 
\°/0.  His  commission  was  $15.  For  how  much  was  the 
house  insured  ? 

20.  A  gold-digger  who  had  3  pounds  of  gold  dust,  lost  9 
ounces.     What  per  cent  was  left  ? 

498.   "Written  Problems. 

1.  What  number  must  be  added  to  the  sum  of  -|,  -J,  and 
\\  to  make  5^? 

2.  Find  the  interest  on  $  2320,  for  5  months  and  21  days, 
at  the  rate  of  7  per  cent  a  year. 

3.  Find  the  interest  on  $640,  from  Sept.  3,  1904,  to 
Oct.  30,  1905,  at  6  per  cent  per  annum. 


386  Chapter  Six. 

4.  At  compound  interest,  what  will  $  200  amount  to  in 
1  year  and  3  months,  at  6  per  cent,  interest  compounded 
semi-annually  ? 

5.  A  man  drew  out  of  the  bank  f  of  his  money,  and  ex- 
pended 30%  of  50%  of  this  for  936  bushels  of  wheat  at 
$  0.87J  a  bushel.     What  sum  had  he  left  in  bank  ? 

6.  A  house  that  cost  $  14,500  rents  for  $  1189.  What 
per  cent  does  it  pay  on  the  investment  ? 

7.  If  4  men  dig  a  ditch  24  rods  long  in  20  days,  how 
long  a  ditch  can  5  men  dig  in  8  days  ? 

8.  For  what  sum  must  a  60-day  note  be  written  to  yield 
$  294.75  at  a  bank,  discounting  at  6%  ? 

9.  An  agent  receives  $  5616  for  silk  he  has  purchased 
and  his  commission  on  it  at  4%.  How  many  yards  did  he 
purchase  at  $  1.50  per  yard  ? 

10.  What  will  be  the  proceeds  of  a  60-day  note  for  $  500, 
dated  June  4,  1904,  and  discounted  at  a  bank  July  1,  1904, 
at  6%  ? 

11.  At  what  rate  will  $142  gain  $21.30  interest  in  3 
years  ? 

12.  What  is  the  duty,  at  50^  a  pound  and  30%  ad  valorem, 
on  700  yards  of  French  broadcloth,  invoiced  at  $1.25  per 
yard,  and  weighing  1  \  pounds  per  yard  ? 

13.  What  will  be  the  amount,  at  compound  interest,  of 
$  340,  at  8%,  for  1  yr.  3  mo.,  the  interest  compounded  semi- 
annually ? 

14.  If  I  lose  10%  by  selling  goods  at  18^  a  yard,  for 
what  must  they  be  sold  to  gain  20%  ? 

15.  I  sold  24  J  %  of  my  estate,  or  $  1372  worth.  I  am 
worth,  in  addition  to  my  real  estate,  $14,000.  How  much 
am  I  worth  in  all  ? 


Fractions.  387 

REVIEW  OF  FRACTIONS. 

499.  Oral  Exercises. 
Give  products : 

1.  84  x  24  =  25  times  84  -  84  =  2100  -  84. 

2.  48x24.                  4.    48x49.  6.   84x74. 

3.  24x36.                   5.    84x49.  7.   48x74. 

8.  84  x  241  =  25  times  84  -  £  of  84  =  2100  -  42. 

9.  48x241               11.   48x24f.  13.   48  x  24f 
10.   36  x  241               12.   36  x  24f.  14.   36  x  24|. 

15.  48  x  361  =  371.  times  48  -  48  =  (|  of  4800)  -  48. 

16.  48xllJ.              17.   48x86£.  18.   48  x  37f 

Give  quotients : 

1.  36  -*-}.                    7.   18I-T-3.  13.   12}-* If 

2.  36-i-|.                     8.    20£-r-4.  14.    16£-*-lf 

3.  36-2}.                   9.    17!- -5- 5.  15.    13^-j-  3f 

4.  36  -i-  f                   10.    191 -J- 6.  16.    14}  + If 

5.  36  -s-lf                11.   16J-5-7.  17.    15}  -j-2f 

6.  36-r-lJ.                 12.    17J-J-8.  18.   17£-i-3f 

500.  Written  Exercises. 
Find  products : 

1.  648  x}.  9.     792x25. 

2.  976  x  if  10.     457x16. 

3.  1648  x87f     .  11.    1864x250. 

4.  2592  x9if  12.      983x51. 

5.  2416x875.  13.   1576  x  62f 

6.  874x99.  14.     176x23f. 

7.  848x125.  15.   1128x875. 

8.  375x999.  16.     895  x  44f 


388  Chapter  Six. 

501.   Written  Exercises. 

1.  Divide  the  sum  of  6f  and  1-|  by  the  difference  be- 
tween 2\  and  3£. 

2.  What  is  the  difference  between  the  sum  of  %  and  | 
and  the  product  of  f  and  fa  ? 

3.  What  is  the  product  of  the  sum  and  the  difference  of 
4|  and  6J  ? 

4.  Subtract  §  of  J  from  fj ;  and  find  the  value  of  fa  of 
16s.  6cZ. 

5.  Add  7f ,  }  of  ft,  I  of  7| ,  and  «. 

6.  Keduce  f  of  a  square  rod  to  the  fraction  of  an  acre, 
and  find  the  value  of  ^  of  a  ton  in  pounds  and  ounces. 

7.  Keduce  T6^^  to  its  lowest  terms,  and  Q*  "~0t  to  its 
simplest  form.  ¥  +    s 

8.  Add  |j  f,  f,  and  J;  multiply  the  sum  by  fa;  and 
subtract  the  product  from  1. 

9.  Find  the  value  of  9^  meters  at  4-J  francs  per  meter. 

10.  Divide  2\  by  3£,  and  add  the  quotient  to  fa. 

11.  Multiply  2  fa  by  16f ,  and  divide  the  result  by  1\  of  2f 

12.  Keduce  7s.  6c?.  to  the  fraction  of  a  pound,  and  7  hr.  12 
min.  to  the  fraction  of  a  day. 

13.  Keduce  to  its  simplest  form  -  +  *      5^. 

14.  Add  together  £  £  and  ^  of  5|  shillings. 

15.  What  fractional  part  of  7  A.  127  sq.  rd.  is  5  A. 
81  sq.  rd.  ? 

16.  What  must  be  added  to  f  of  j-  to  make  it  equal  to 
AofSf? 

17.  \  of  a  number  is  148.     What  is  the  number  ? 

18.  If  4  of  a  field  is  worth  $325,  what  is  the  field  worth  ? 

19.  If  J  of  a  house  is  worth  $4900,  what  is  the  value 
of  i? 


Denominate  Numbers.  389 

REVIEW  OF  DENOMINATE  NUMBERS. 

502.  Written  Exercises. 

1.  Change  43  yards  to  rods  and  a  fraction. 

2.  Change  43  yards  to  rods  and  yards. 

43  yards  -4-  5|  yards  gives  the  number  of  rods. 

3.  Change  43  yards  to  rods,  yards,  and  feet. 

4.  Change  43  yards  to  rods,  yards,  feet,  and  inches. 

5.  Change  72  yards  to  rods,  etc. 

6.  Change  66  yards  to  rods. 

Change  to  rods,  yards,  etc. : 

7.  49  yards.  11.  1836  inches. 

8.  147  feet.  12.  1837  inches. 

9.  1764  inches.  13.  52  yards. 
10.    8Jf  rods.  14.  49J  yards. 

503.  Change  to  rods,  etc. : 

15.  1483  inches.       18.   2796  inches.       21.   3453  inches. 

16.  984  inches.        19.    1121  inches.        22.    1278  inches. 

17.  1345  inches.       20.    1470  inches.       23.   1576  inches. 

504.  Add: 

24.   4  rd.  3  yd.  1  ft.  25.    5  rd.  4  yd.  2  ft. 

9  rd.  4  yd.  2  ft.  5  yd.  1  ft. 

3  rd.  1  ft.  6  in.  6  rd.  1  yd. 

26.  From  8  rd.  1  ft.  take  2  rd.  2  ft. 

27.  Find  the  difference  between  3  rd.  1  yd.  1  ft.  and  16  rd. 

28.  Multiply  5  rd.  4  yd.  2  ft.  by  4. 

29.  Multiply  11  rd.  2  ft.  by  10. 

30.  Divide  30  rd.  5  yd.  2  ft.  by  8. 

31.  Divide  34  rd.  2  yd.  by  9. 


390  Chapter  Six. 

REVIEW  OF  COMMERCIAL  DISCOUNT. 

505.  Oral  Exercises. 

When  the  list  price  is  $  1,  what  is  the  net  price  after  the 
deduction  of  each  of  the  following  discounts  ? 

1.  30  and  20%.  4.   50  and  10%. 
The  net  price  after  a  deduction  ~     ^q       j  2Q  ol 

of  30  %  is  70  %  of  .$  1 ,  or  70  f.    De- 
ducting 20  %  of  70  f  leaves  80  %  of  6.10  and  5  % . 

W'or66^-  7.   20  and  20%.  • 

2.  40  and  10%.  m     001       ,  1|A-I 

7  8.   334  and  10%. 

The  net  price  is  G0%  of  90%  of  6 

$1,  or  54%  of  |1.  9.    20  and  15%. 

3.  25  and  40%.  10.   30  and  15%. 

506.  What  single  discount  is  equal  to  each  of  the  follow- 
ing double  discounts  ? 

11.  30  and  30%.  15.   40  and  30%. 

The  net  price  is  70%  of  70%  of  ^     ^  ^  ^ 

list  price,  or  40%  of  list  price. 

The  discount  is,  therefore,  100%  17,    40  and  5%. 

-49%,  or  51%. 

™       -,  ***  18-   50  and  20%. 

12.  20  and  25%.  ' 

13.  25  and  20%.  19'   40  and  15%. 

14.  15  and  30%.  20.    50  and  15%. 

Find  the  single  discount  equal  to  each  of  the  following : 

21.  50  and  20  and  10%. 

22.  40  and  25  and  20%. 

23.  10  and  10  and  10%. 

24.  30  and  20  and  10%. 


Interest.  391 

507.  Written  Exercises. 

Which  is  the  better  discount  for  the  buyer  ? 

1.  30  and  20%,  or  40  and  10%. 

30  and  20%  off  =  70%  of  80%  net,  or  56%  net.  40  and  10%  off  s 
60%  of  90%  net,  or  64%  net.     The  latter  is  the  better  for  the  buyer. 

2.  50  and  10%,  or  40  and  20%. 

3.  20  and  20%,  or  30  and  10%. 

4.  20  and  15%,  or  30  and  5%. 

5.  30  and  15%,  or  25  and  20%. 

6.  30  and  30%,  or  50  and  10%. 

7.  40  and  30%,  or  20  and  50%. 

8.  40  and  5%,  or  30  and  15%?. 

9.  20  and  50%,  or  60  and  10%. 
10.   40  and  15%,  or  30  and  25%. 

REVIEW  OF  INTEREST. 

508.  Six  Per  Cent  Method. 

Interest  is  the  product  of  the  principal  by  the  rate  ex- 
pressed as  hundredths  by  the  time  in  years  and  fraction. 
The  usual  method  is  to  perform  the  operations  in  the  above 
order.  When  the  rate  is  6%,  some  prefer  to  first  multiply 
the  rate  by  the  time,  and  to  use  this  as  a  multiplier  of  the 
principal. 

In  finding  the  product  of  the  rate  by  the  time,  advantage 
is  taken  of  the  fact  that  6  is  a  factor  of  12  and  30.  Six  per 
cent  a  year  is  -J  per  cent  a  month  and  -fa  per  cent  a  day. 

When  the  rate  is  a  different  per  cent,  the  interest  is  first 
obtained  at  6  per  cent  by  this  method,  and  from  this  result 
the  interest  is  calculated  for  the  given  rate. 


39* 


Chapter  Six. 


509.   Find  the 
7  mo.  19  da. 


$2874.35 


•218* 


.47905+ 
22.99480 
28.7435 
574.870 
6)$  627.08735 
$104.5145+ 

3f 

313.5435 
78.3859 


$391.9294 
$391.93  Ans. 


interest  on  $2874.35  at  3f%  for  3  ft. 

6  %  for  1  year  is  for  3  years  .18 

\  %  for  1  month  is  for  7  months  .035 

BV/0  for  1  day  is  for  19  days  .003$ 

6  %  for  3  yr.  7  mo.  19  da.  is  .218  \ 

Multiplying  the  principal  by  .218$  gives  the 
interest  at  6  %  for  3  yr.  7  mo.  19  da. 

Dividing  this  product  by  6  gives  the  interest 
at  1  %.  Multiplying  the  quotient  by  3f  gives  the 
interest  at  3f  %. 

To  find  the  interest  at  6  per  cent,  multi- 
ply the  principal  by  6  times  the  number  of 
years  and  J  the  number  of  months  as  hun- 
dredths, together  with  %  the  number  of  days 
as  thousandths. 


510.   Written  Exercises. 
Find  the  interest  at  6%  on: 

1.  $  1428  for  1  yr.  4  mo.  6  da. 

2.  $  372.50  for  2  yr.  6  mo.  24  da. 

3.  $  1875  for  3  yr.  9  mo.  18  da. 

4.  $  240  for  4  yr.  7  mo.  15  da. 

5.  $  92.75  for  5  yr.  4  mo.  8  da. 

6.  $  817.80  for  10  mo.  19  da. 

The  interest  at  6  %  plus  $  of  itself  gives  the  interest  at  7  %. 
The  interest  at  6  %  minus  $  of  itself  gives  the  interest  at  5  %. 
The  interest  at  6  %  plus  $  of  itself  gives  the  interest  at  8  %. 
The  interest  at  6  %  minus  $  of  itself  gives  the  interest  at  4  %. 
The  interest  at  6  %  plus  \  of  itself  gives  the  interest  at  1\  %. 
The  interest  at  6  %  minus  \  of  itself  gives  the  interest  at  4 \  %. 


Interest.  393 

511.   "Written  Exercises. 
Find  the  amount : 

1.  $1875.25  for  3  yr.  5  mo.  15  da.,  at  4|%. 

2.  $487.50  for  1  yr.  10  mo.  25  da.,  at  6%. 

3.  $1206.84  for  2  yr.  1  mo.  16  da.,  at  5%. 

4.  $595.00  for  7  yr.  7  mo.  7  da.,  at  7%. 

5.  $763.25  for  8  mo.  11  da.,  at  4%. 

6.  $685.70  for  19  mo.  5  da.,  at  51%. 

7.  $1563.00  for  3  mo.  20  da.,  at  5%. 

8.  $998.45  for  87  da.,  at  4|%. 

9.  $2575.50  for  149  da.,  at  3%. 

10.  $693.27  for  214  da.,  at  2£%. 

Find  the  principal,  rate,  or  time : 

11.  Principal,  $240;  interest,  $32.04;  time,  2  yr.  11  mo. 

18  da.    Eate  ? 

12.  Eate,  6%  ;  amount,  $717.40;  time,  3  yr.  3  mo.  4  da. 
Principal  ? 

13.  Principal,  $360;  rate,  3%  ;  interest,  $48.87.    Time? 

14.  Principal,  $288;  rate,  2£%;  amount,  $307.22.   Time? 

15.  Eate,   6%;    interest,   $13.10;    time,    4   mo.   11   da. 
Principal  ? 

16.  Principal,    $270;   amount,    $273.27;   time,    3    mo. 

19  da.     Eate  ? 

17.  Eate, 4£% ;  interest,  $  25.11 ;  principal,  $ 360.   Time? 

18.  Interest,  $50.22;  time,  3  yr.  1  mo.  6  da. ;  rate, 
Amount  ? 


394  Chapter  Six. 

REVIEW  OF  BANK  DISCOUNT. 

512.  Written  Exercises. 

Find  face  of  note,  term  of  discount,  rate,  discount,  or 
proceeds : 

By  the  term  is  meant  the  number  of  days  the  note  has  to  run, 
including  grace,  if  any. 

1.  Face,  $600;  discount,  $6.30;  rate,  6%.     Term? 

2.  Term,  33  days;  proceeds,  $397.80;  rate,  6%.     Face? 

3.  Term,  90  days;  face,  $300;  rate,  6%.     Proceeds? 

4.  Term,  21  days;  face,  $600;  discount,  $2.45.     Rate? 

5.  Term,  4  months;  face,  $200;  rate,  6%.     Discount? 

6.  Term,  132  days ;  proceeds,  $  2689.50 ;  rate,  6 % .    Face ? 

7.  Face,  $150;  proceeds,  $147.75;  rate,  6%.     Term? 

8.  Face,  $1650;  discount,  $4.95;  rate,  6%.     Term? 

9.  Term,  69  days;  proceeds,  $469.30;  rate,  6%.    Face? 

EXACT  INTEREST. 

Exact  interest  is  used  by  the  United  States  Government  in  its  cal- 
culations.    365  days  are  taken  to  the  year. 

513.  Written  Exercises. 

1.  Find  the  exact  interest  of  $280  from  April  14  to 
Sept.  6  at  4%. 

Time,  145  days.     Ans.  $280  xT^x  }f$. 

2.  Find  the  exact  interest  on  $76.65  from  March  4  to 
Dec.  15  at  6  per  cent. 

3.  On  $  384  at  7^  per  cent  for  75  days. 

4.  On  $438  at  5%  from  Jan.  1  to  March  15. 

5.  On  $109.50  at  4|%  for  87  days. 

6.  On  $847.60  at  5%  from  April  29  to  Sept.  20. 

7.  $584  at  3J%  from  May  16  to  Dec.  1. 

Unless  "exact"  or  "accurate"  interest  is  specified,  use  360  days 
to  the  year. 


Review.  395 

MISCELLANEOUS. 

514.    Oral  Review  Problems. 

1.  A  has  96  sheep ;  B  has  28  sheep  more  than  A.  How 
many  sheep  have  both  ? 

2.  There  are  56  pupils  in  one  class,  48  in  a  second  class, 
and  52  in  a  third  class.  How  many  pupils  are  there  in  the 
three  classes  ? 

3.  March  29  is  what  day  of  the  year  1904  ? 

4.  How  far  is  a  man  from  his  starting-point,  if  he  travels 
due  east  150  miles,  due  west  23  miles,  due  east  again  48 
miles  ? 

5.  A  body  falls  16  feet  in  the  first  second,  three  times 
as  far  in  the  second  second,  five  times  as  far  in  the  third 
second.     How  far  does  it  fall  in  three  seconds? 

6.  The  base  of  a  right-angled  triangle  is  12  feet,  the 
perpendicular  is  16  feet.     What  is  the  hypotenuse  ? 

7.  At  $35  per  month,  what  will  be  the  rent  of  a  house 
for  16  months  ? 

8.  A  field  containing  169  square  rods  is  13  rods  long. 
What  is  the  perimeter  ? 

9.  25  packages  of  sugar  weigh  together  87£  pounds. 
How  many  pounds  are  there  in  each  ? 

10.  At  45  miles  per  hour,  how  many  hours,  minutes,  and 
seconds  will  it  take  a  train  to  go  230  miles  ? 

11.  How  many  years  have  elapsed  since  the  invention  of 
gunpowder,  1356  ? 

12.  What  profit  is  made  on  an  article  bought  for  $175, 
less  12%,  and  sold  for  $200  ? 

13.  How  many  square  rods  in  a  field  71  rods  long,  81 
rods  wide  ? 


396  Chapter  Six. 

14.  Assuming  a  kilo  to  be  2\  pounds,  how  many  kilos 
will  be  equal  to  143  pounds  ? 

15.  A  degree  of  longitude  in  latitude  45°  is  about  70%  of 
the  length  of  a  degree  on  the  equator.  Calling  the  latter 
length  69  miles,  how  long  is  a  degree  of  longitude  in  latitude 
45°? 

16.  At  $44  per  acre,  how  much  land  can  be  bought  for 


17.  A  number  of  marbles  divide'd  among  29  boys  gives 
each  16  marbles,  and  leaves  a  remainder  of  26.  How  many 
marbles  are  there  ? 

18.  What  is  the  monthly  salary  of  a  clerk  who  receives 
$1500  per  year? 

19.  How  many  revolutions  in  a  mile,  5280  feet,  are  made 
by  a  locomotive  wheel  16  feet  in  circumference  ? 

20.  What  is  the  perimeter  of  a  lot  49  feet  wide,  87  feet 
long? 

21.  How  many  bricks  8  inches  by  4  inches  by  2  inches 
would  make  a  cubic  foot  ? 

22.  13  is  one  factor  of  1001.  Find  the  other  two  prime 
factors. 

23.  What  are  the  three  equal  factors  of  343  ? 

24.  What  is  the  square  root  of  1225  ? 

25.  At  4 J  miles  per  hour,  how  long  will  it  take  a  man  to 
walk  37£  miles  ? 

26.  What  will  be  the  cost  of  9  dozen  hats  at  $1.33^ 
each? 

27.  Paid  92^  for  coffee,  48^  for  butter,  and  18^  for  lard. 
How  much  was  my  bill  ? 

28.  I  had  $  150.  Spent  $  23  for  a  suit  of  clothes  and  $  48 
for  tools.     How  much  was  left  ? 

29.  What  is  the  area  of  a  field  36  yards  by  31  yards  ? 


Review.  397 

30.  600  hours  equal  how  many  days  ? 

31.  What  is  the  cost  of  a  cow  if  I  pay  $630  for  15? 

32.  How  many  ounces  in  2&±-  pounds  ? 

33.  109J  pounds  of  sugar  are  divided  among  4  people. 
What  is  the  share  of  each  ? 

34.  At  1T97^  per  pound,  how  many  pounds  of  iron  can  1 
get  for  $ '5.70  ? 

35.  What  is  the  cost  of  51  tons  iron  at  $  17  per  ton  ? 

36.  What  will  be  the  average  age  of  9  boys,  each  12  years 
old,  and  6  boys,  each  10  years  old  ? 

37.  At  42  miles  per  hour,  how  long  will  it  take  a  train  to 
go  882  miles  ? 

38.  At  25^  per  hour,  what  will  a  man  earn  in  18  days  of 
10  hours  ? 

39.  What  will  ^be  the  net  price  of  an  article  whose  cata- 
logue price  is  $20.00,  the  discount  being  90  and  10%  ? 

40.  A  man  had  $  181  in  bank.    What  will  be  his  balance 
after  taking  out  $47  and  $33  ? 

41.  How  many  feet  in  14  rods  ? 

42.  77  yards  are  how  many  rods  ? 

43.  How  many  square  yards  are  there  in  a  floor  lOf  yards 
long  and  6^  yards  wide  ? 

44.  What  is  the  cost  of  372  eggs  at  15^  per  dozen  ? 

45.  A  man  owns  3  farms  containing  65  acres,  86  acres,  and 
98  acres,  respectively.     How  many  acres  does  he  own  ? 

46.  What  is  the  area  of  a  piece  of  glass  measuring  8J  by 
6J  inches  ? 

47.  What  is  the  value  in  U.  S.  money  of  50  marks  at  23^ 
cents  ? 

48.  How  many  francs  will  a  calf  cost,  if  18  are  worth  630 
francs  ? 


398  Chapter  Six. 

49.  A  man  spends  $  1740  per  year.  What  is  the  average 
amount  spent  per  month  ? 

50.  What  would  51  pounds  of  butter  cost  at  33^  ^  a  pound  ? 

51.  Mrs.  Allen  bought  7  chairs  at  $4  apiece,  2  tables  at 
$9  apiece,  and  a  carpet  for  $33.  She  paid  two  f  50  bills. 
How  much  change  was  due  her  ? 

52.  In  what  time  will  any  sum  of  money  double  itself,  at 
6%? 

53.  Find  the  sum  of  the  prime  numbers  as  far  as  12. 

54.  Interest  of  $1234,  for  30  days,  at  6%  ? 

55.  Interest  of  f  1234,  for  6  months,  at  4%  ? 

56.  Oil  is  worth  37^  a  pint.  How  many  pints  can  be 
bought  for  $6? 

57.  Sold  oranges  for  \$  apiece,  gaining  50%.  How  much 
did  they  cost  apiece  ? 

58.  What  will  be  the  cost  of  1  pk.  1  qt.  1  pt.  of  nuts,  at 
10  ^  per  quart  ? 

59.  What  is  the  value  of  an  acre  of  land,  at  10^  per 
square  foot  ? 

60.  3  desks  are  bought  at  $  10  each,  and  sold  for  $  45. 
Find  the  rate  of  gain. 

61.  A  wheelman  sells  his  old  bicycle  for  $25,  and  loses 
16}%.     How  much  did  it  cost  him  ? 

62.  How  much  does  an  agent  get  for  buying  5  bale?  of 
goods  at  $  400  each,  if  he  receives  3%  for  his  services  ? 

63.  10%  of  200  is  £  of  what  number  ? 

64.  How  old,  December  1, 1903,  was  a  boy  born  November 
25,1889? 

65.  A  man  has  $  1000  in  bank.  What  will  remain  after 
he  has  taken  out  $  478  ? 

66.  How  many  hours  in  the  month  of  January  ? 


Review.  399 

67.  In  how  many  years,  months,  and  days  will  $100 
amount  to  $111,  at  5%,  simple  interest? 

68.  What  will  5  tons  of  granulated  sugar  cost,  at  6 J  ^  per 
pound  ? 

69.  What  is  the  interest  of  $  50,  for  3  yr.  7  mo.  12  da.,  at 
6%? 

70.  A  farmer  makes  675  gallons  of  cider.  He  has  but  12 
barrels,  each  of  45  gallons'  capacity,  to  store  it  in.  How 
many  more  such  barrels  does  he  need  ? 

71.  What  will  be  the  cost  of  36  yards  of  cloth,  at  $2.75 
per  yard  ? 

72.  Add  3794  and  2975. 

73.  What  is  the  bank  discount  on  a  sixty-days  note  for 
$400,  at  6%  ? 

74.  Change  ^-toa  decimal  of  three  places. 

75.  How  much  wood  in  three  piles  containing,  respec- 
tively, \  of  a  cord,  ^  of  a  cord,  and  \  of  a  cord  ? 

76.  What  is  the  percentage  of  gain  in  case  of  railroad 
stock  bought  for  $  80  per  share,  and  sold  for  $  90  per 
share  ? 

77.  A  dealer  sold  flour  at  a  profit  of  50^  a  barrel,  and 
gained  10%.     What  was  the  cost  ? 

78.  At  10  ^  a  quart,  what  are  3  bu.  1  pk.  5  qt.  of  chest- 
nuts worth  ? 

79.  How  many  yards  in  288  inches  ? 

80.  What  decimal  of  a  number  is  -f  per  cent  of  it  ? 

81.  If  a  broker  buys  for  me  5  shares  of  railroad  stock 
whose  par  value  is  $  100,  what  is  his  brokerage  at  \%  ? 

82.  If  I  sell  10  shares  of  railroad  stock  for  $  1090,  and 
gain  9%  on  the  cost,  what  was  the  cost? 

83.  What  is  the  interest  of  $  660,  for  3  months,  at  4%  ? 


4-00  Chapter  Six. 

84.  What  per  cent  does  a  merchant  lose  by  selling  goods 
at  |  of  their  cost  ? 

85.  What  principal  at  6%  simple  interest  will  gain  $36 
in  1  year  and  6  months  ? 

86.  What  per  cent  is  gained  on  goods  sold  at  double  the 
cost? 

87.  What  is  8%  of  50  bushels  ? 

88.  $3000  is  11£%  of  my  property.  How  much  am  I 
worth  ? 

89.  What  is  the  interest  on  $  700,  for  15  days,  at  6%  ? 

90.  Bank  discount  on  a  65-day s  note  for  $  1000,  dis- 
counted at  date  ? 

91.  At  what  rate  will  $  2  gain  $  20  in  5  years  ? 

92.  A  capitalist  wishes  to  realize  5%  on  money  invested 
in  stock.  What  must  be  the  annual  dividend  on  stock  cost- 
ing 300,  in  order  to  produce  this  rate  ? 

93.  What  will  be  the  taxes  on  property  assessed  at 
$25,000,  the  rate  being  $16  per  $1000  ? 

94.  Find  the  compound  interest  on  $  1000,  for  two  years, 
at  five  per  cent,  interest  compounded  annually. 

95.  What  will  be  the  net  cost  of  an  article  marked  $8, 
on  which  a  discount  of  50,  25,  and  10%  is  allowed  ? 

96.  Find  the  "list"  price  of  an  article  sold  for  $  10  after 
a  discount  of  50  and  50  per  cent  had  been  deducted. 

97.  Paid  90  ^  for  an  article.  The  discount  is  25  and  25 
per  cent.     What  is  the  list  price  ? 

98.  One  boy  can  do  a  certain  piece  of  work  in  2  hours,  a 
second  boy  requires  3  hours,  a  third  needs  6  hours.  How 
long  will  it  take  the  three  working  together  ? 

99.  Sold  a  cow  for  $  60,  losing  25  %  •    What  was  the  loss  ? 
100.   Sold  a  cow  for  $60,  gaining  25%.     What  was  the 

gain? 


Review.  401 

101.  Sold  two  horses  at  $240  apiece.  On  one  I  gained 
20%,  on  the  other  I  lost  20%.  Did  I  gain  or  lose  on  both, 
and  how  much  ? 

102.  What  is  the  interest  of  $1500,  for  60  days,  at  6%  ? 

103.  How  many  years  will  it  take  $20  to  gain  $20  at 
5  per  cent  simple  interest  ? 

104.  John  has  $60,  James  has  $80.  James  has  what 
per  cent  more  money  than  John  ?  John  has  what  per  cent 
less  money  than  James  ? 

105.  I  is  what  per  cent  of  J  ?     J  is  what  per  cent  of  f  ? 

106.  Two  men  working  together  can  finish  a  piece  of 
work  in  8  days ;  one  can  do  it  in  12  days.  How  long  would 
the  other  take  to  do  the  work  ? 

107.  How  many  yards  of  cloth  at  $3.75  per  yard  can  be 
bought  for  $90? 

108.  A  puts  $600  into  business;  B,  $400;  the  profits 
are  $500.     What  is  the  share  of  each  ? 

109.  Two  boys  hire  a  camera  for  26  weeks,  paying  $5.20. 
How  much  should  be  paid  by  the  boy  that  uses  it  12  weeks  ? 

110.  New  Orleans  is  90°  west  of  Greenwich.  When  it  is 
2  p.m.  at  the  latter  place,  what  is  the  time  at  New  Orleans  ? 

111.  Find  the  discount,  at  6%,  on  a  note  for  $300,  that 
has  48  days  to  run. 

112.  What  will  be  the  cost  of  84  yards  of  cloth  at  49  ? 
a  yard  ? 

113.  Two  men  hire  a  pasture  for  $84.  One  puts  in  twice 
as  many  head  of  cattle  as  the  other.    What  should  each  pay? 

114.  A  base-ball  club  won  17  games,  and  lost  13  games. 
What  per  cent  of  its  games  did  it  win  ? 

115.  What  per  cent  of  4  is  64  ? 

116.  2|  is  what  per  cent  of  3£? 

117.  How  many  acres  in  a  rectangular  farm  1  mile  long, 
J  mile  wide  ? 


402  Chapter  Six. 

118.  What  per  cent  of  the  "list"  price  is  paid  by  a  buyer 
who  receives  a  discount  of  20  and  10  per  cent  ? 

119.  A  tank  is  filled  by  two  pipes,  one  of  which  can  fill 
it  in  6  hours,  and  the  other  in  8.  How  long  will  it  take 
both  together  to  fill  the  tank  ? 

120.  Find  the  interest  on  $80,  for  72  days,  at  6%. 

121.  A  man  sold  a  wagon  for  $420,  which  was  16%  less 
than  it  cost.     How  much  did  he  lose  ? 

122.  A  kilo  is  2.2046  lb.  How  many  pounds  in  1000 
kilos  ? 

515.  "Written  Keview  Problems. 

1.  What  number  subtracted  88  times  from  80.005  will 
leave  .013  as  a  remainder  ? 

2.  At  what  price  must  an  article  that  cost  $30  be 
marked  so  that  after  deducting  40%  from  the  marked  price, 
30  %  profit  may  be  realized  ? 

3.  Write  a  ninety-days  promissory  note  for  which  you 
should  get  $  240  at  the  bank,  discount  being  6%. 

4.  If  a  horse  dealer  buys  a  span  of  horses  at  10  per  cent 
less  than  their  value,  and  sells  them  at  10  per  cent  more 
than  their  value,  what  per  cent  does  he  make  ? 

5.  If  a  boy  buys  peaches  at  the  rate  of  5  for  2  cents,  and 
sells  them  at  the  rate  of  4  for  3  cents,  how  many  must  he 
buy  and  sell  to  gain  $  4.20  ? 

6.  What  is  the  difference  between  the  compound  interest 
on  $  5000,  for  3  years,  at  5%,  and  on  $  10,000,  for  1£  years, 
at  the  same  rate  ? 

7.  A  can  do  a  piece  of  work  in  27  days,  and  B  in  15 
days ;  A  works  at  it  alone  for  12  days,  B  then  works  alone 
for  5  days,  then  C  finishes  the  work  in  4  days.  In  what 
time  could  C  have  done  the  work  by  himself  ? 


Review.  403 

8.  A  room  is  15  feet  long,  10  feet  broad,  and  9  feet  9 
inches  high.  Find  the  cost  of  painting  the  walls  and  the 
ceiling,  at  Is.  9d.  a  square  yard. 

9.  What  is  the  value  of  a  pile  of  wood  40  feet  long,  4 
feet  wide,  and  5  feet  high,  at  $  5.30  a  cord? 

10.  By  buying  a  cargo  of  coal  at  $6  per  ton,  and  selling 
it  at  $  8  a  ton,  I  gained  $  198.     How  much  did  I  pay  for  it  ? 

11.  Make  out  a  receipted  bill  for  the  following :  325  yards 
of  silk  at  $  2.25  per  yard ;  296  yards  of  lace  at  $  1.50  per 
yard ;  480  yards  of  ribbon  at  9  0.50  per  yard ;  45  dozen 
pairs  of  gloves  at  $  15  per  dozen  pairs. 

12.  My  dividend  is  8|,  quotient  94.     What  is  the  divisor  ? 

13.  I  gave  away  ^  and  -§  of  41  bushels  of  chestnuts. 
What  %  was  left  ? 

14.  The  perimeter  of  a  square  field  is  16  rods.  WTiat  is 
the  field  worth,  at  8J^  a  square  foot? 

15.  A  broker's  bill  for  cotton  at  4J^  per  pound  and  his 
commission  for  buying  at  21%  was  $1998.75.  How  many 
bales  of  400  pounds  each  did  he  buy,  and  what  was  his 
commission  ? 

16.  I  sold  80  yards  of  broadcloth  for  $  240,  thereby  losing 
20%  on  the  cost.  For  what  should  I  have  sold  it  per  yard 
to  have  gained  15  %  on  the  cost  ? 

17.  A  man  bought  60  casks,  of  65  gallons  each,  for 
$  1542;  80  gallons  leaked  out.  For  what  must  he  sell  the 
remainder  per  gallon  to  gain  12  J  %  on  the  cost? 

18.  Each  of  two  men  sold  his  horse  for  $  180.  One  made 
20%,  the  other  lost  20%  on  the  cost.     Cost  of  each  horse? 

19.  A  man  agrees  to  dig  a  cellar  30  feet  long,  24  feet 
wide,  and  6  feet  deep.  What  %  of  the  work  is  to  be  done 
when  he  has  removed  144  cubic  yards  ? 


404  Chapter  Six. 

20.  A  man  bought  672  yards  of  cloth  at  $  1.25  a  yard. 
He  sold  it  immediately  for  $  2.25  a  yard,  receiving  in  pay- 
ment a  60-day s  note  for  the  amount,  which  he  had  dis- 
counted at  a  bank  at  7%.     How  much  money  did  he  make  ? 

21.  What  will  it  cost  to  fill  in  a  street  55  feet  wide, 
600  feet  long,  and  5|  feet  below  grade,  at  40  ^  a  cubic  yard  ? 

22.  The  quotient  arising  from  the  division  of  6985.473  by 
a  certain  number  is  51,  and  the  remainder  is  68.853.  What 
is  the  divisor  ? 

23.  What  is  the  value  of  the  following? 

8flr  +  **-«t  .1  xftxf 
2f  +  l|-3i   "   fxfxi 

24.  In  going  1  mi.  94  rd.  2  yd.  1  ft.,  a  carriage  wheel 
makes  526  revolutions.  What  is  the  circumference  of  the 
wheel  ? 

25.  On  a  note  dated  Oct.  16,  1903,  for  $2650,  with  in- 
terest at  6  per  cent,  the  following  payments  were  made: 
Jan.  28, 1904,  $  575 ;  May  22, 1904,  $  25 ;  and  Aug.  4,  1904, 
$  948.    'What  was  due  Nov.  25,  1904  ? 

26.  A  grocer  pays  18  ^  per  pound  for  coffee  and  roasts  it, 
the  coffee  losing  10  per  cent  of  its  weight  in  the  process. 
What  must  he  charge  per  pound  for  the  roasted  coffee  in 
order  to  make  a  profit  of  20  per  cent,  allowing  4  per  cent 
for  bad  debts  ? 

Note.  —  96%  of  the  price  he  receives  per  pound  must  be  20%  more 
than  the  rate  of  18  f  for  ^  lb. 

27.  A  merchant  imported  from  Bremen  32  pieces  of  linen 
of  32  yards  each,  on  which  he  paid  for  the  duties,  at  24  per 
cent,  $  122.16,  and  other  charges  to  the  amount  of  $  40.96. 
What  was  the  invoice  value  per  yard,  and  the  cost  per  yard 
after  duties  and  charges  were  paid  ? 


Review.  405 

28.  A  garrison  of  1200  men  is  provisioned  for  100  days. 
At  the  end  of  30  days,  600  men  are  withdrawn,  and  at  the 
end  of  60  days,  900  men  are  added.  How  long  will  the 
provisions  last  ? 

29.  What  will  be  the  result,  if  £  of  f  of  3  J-  be  multiplied 
by  £  of  itself,  and  the  product  be  divided  by  £  ? 

30.  A  collector  of  internal  revenue  deposited  in  the  treas- 
ury $  762,742.50,  retaining  2\  per  cent  of  the  amount  col- 
lected.    What  amount  did  he  collect  ? 

31.  What  is  the  duty  on  25  tons  2  cwt.  3  qr.  of  iron  at 
%  8  per  ton  ?     (1  ton  =  2240  lb.) 

32.  An  importer  sold  a  part  of  a  cargo  of  tea  at  30  cents 
a  pound  and  made  a  profit  of  20  per  cent.  What  per  cent 
did  he  make  on  the  remainder  of  the  cargo,  which  he  sold  at 
40  cents  a  pound  ? 

33.  Divide  %  4.14  among  Thomas,  Richard,  and  Henry  in 
such  a  way  that  Henry  shall  receive  3  cents  for  every  5 
cents  that  Thomas  gets,  and  Richard  shall  receive  2  cents 
for  every  5  cents  that  Henry  gets. 

34.  Reduce  272  liquid  quarts  to  dry  quarts. 

35.  A  pipe  discharging  3  gallons  1  pint  a  minute  fills  a 
tub  in  4  minutes  20  seconds.  Another  pipe  discharges  83 
quarts  a  minute.  If  both  pipes  discharge  together  into  the 
tub,  how  long  will  they  take  to  fill  it  ? 

36.  William  Wilson  sold  goods  to  the  amount  of  %  1000. 
One-half  of  his  sales  showed  a  profit  of  25  per  cent  on  the 
cost,  and  the  remaining  half  a  loss  of  16|  per  cent  on  the  cost. 
Required  the  total  cost  of  the  goods. 

37.  If  I  sell  I  of  my  farm  for  f  of  what  the  farm  cost  me, 
what  is  my  gain  per  cent  ? 

38.  Which  is  the  higher  rate  of  freight  on  wheat,  $.16 
per  hundred  or  $  .10  per  bushel  (60  lb.),  and  what  per  cent  ? 


406  Chapter  Six. 

39.  Write  in  words  : 

(a)  .267;    (6)200.067;   (c)  flftj   ((f)  200^fo. 

40.  If  40  per  cent  of  the  selling  price  of  an  article  is 
profit,  what  is  the  per  cent  of  gain  on  the  cost? 

41.  What  number  added  to  4 J  times  itself  will  equal  60  J  ? 

42.  Divide  J  by  .00003J. 

43.  Keduce  to  lowest  terms  (a)  i|-f| ;    (6)  |f|. 

44.  If  4  men  eat  64  pounds  of  bread  in  2  weeks,  how 
many  pounds  will  16  men  eat  in  7  weeks  at  the  same  rate  ? 

45.  Divide  .75  of  17f  by  £  of  .035. 

46.  Find  the  cost  of  3846  pounds  of  hay  at  $15  per  ton. 

47.  Find  the  cost  of  plastering  the  walls  and  the  ceiling 
of  a  hall  72  feet  long,  50  feet  wide,  and  22  feet  high,  at  18f 
•cents  a  square  yard,  allowing  972  square  feet  for  openings 
•and  baseboards. 

48.  A  certain  quantity  of  paper  will  make  4000  copies  of 
an  octavo  book  (8  pages  to  the  sheet).  How  many  copies 
■of  a  12mo  book  (12  pages  to  the  sheet)  will  the  same  paper 
make? 

49.  Find  the  diagonal  of  a  square  park  containing  20  acres. 

50.  Bangor,  Maine,  June  24,  1904. 
On  demand,  I  promise  to  pay  Joseph  I.  Totten,  or  order, 

Two  Thousand  Five  Hundred  Fifteen  Dollars,  with  interest, 
value  received. 

$2515^.  Charles  Hettesheimer. 

$1541.01  was  paid  Jan.  1,  1905.  Find  the  amount  due 
Aug.  15,  1905. 

51.  How  much  does  it  cost  annually  to  insure  the  "Celtic" 
for  $1,525,000,  if  2J%  is  paid  for  the  insurance  ? 

52.  $  150  is  paid  an  agent  for  purchasing  1200  barrels  of 
■flour  on  a  commission  of  2£%.  How  much  was  paid  per 
barrel  for  the  flour  ? 


Review.  407 

53.  An  agent  received  $2562.50  for  purchasing  land  at 
$62.50  per  acre,  and  his  commission  of  2|%.  How  many 
acres  did  he  buy  ? 

54.  Reduce  the  fraction  -3 — 4S  +  « — I*  5  • 

55.  Divide  J  by  2.5,  to  the  quotient  add  the  divisor,  and 
from  that  sum  subtract  the  dividend.  Give  the  fractional 
part  of  the  answer  in  a  decimal. 

56.  If  the  interest  on  $300  for  1  yr.  8  mo.  is  $36,  find 
what  would  be  the  interest  on  $  212.50  for  3  yr.  4  mo.  24  da* 
at  the  same  rate. 

57.  Reduce  .0468  T.  to  a  compound  number. 

58.  Find  the  prime  factors  of  20,930. 

59.  A  man  paid  $  999  for  the  rent  of  a  house  from 
June  29,  1903,  to  May  5,  1905.  What  was  the  rent  per 
year? 

60.  What  per  cent  of  3  lb.  7  oz.  is  7  lb.  9  oz.  ? 

61.  At  50  cents  per  running  yard,  what  will  be  the  cost 
of  fencing  a  square  field  containing  10  acres  ? 

62.  At  the  rate  of  20  problems  an  hour  for  A,  and  15  in 
55  minutes  for  B,  in  what  time  can  both  together  solve  100* 
problems  ? 

63.  Find  the  entire  surface  of  a  cube  whose  edge  meas- 
ures 15  inches. 

64.  A  dealer  buys  books  at  $  1.50  each,  less  33J  and 
10  per  cent.  At  what  price  per  copy  must  he  sell  them 
to  gain  43£  per  cent? 

65.  Abraham  Lincoln  died  at  the  age  of  56  yr.  2  mo.  3  da.,, 
after  serving  as  President  4  yr.  1  mo.  11  da.  Give  the  date 
of  his  birth,  the  date  of  his  inauguration  being  March  4, 
1861. 


408  Chapter  Six. 

66.  A  dealer  buys  150  barrels  of  flour.  He  sells  one- 
third  of  it  at  $  4.50  per  barrel,  losing  10  per  cent.  The 
remainder  he  sells  at  a  profit  of  6  per  cent.  What  is  his  net 
gain  or  loss  ? 

67.  Sixty  per  cent  of  66%  per  cent  of  a  number  equals 
810.     What  is  the  number  ? 

68.  A  ladder  40  feet  long  is  so  placed  in  a  street,  that 
without  being  moved  at  the  foot,  it  will  reach  a  window 
on  one  side  33  feet,  and  on  the  other  side  21  feet  from  the 
ground.     What  is  the  breadth  of  the  street  ? 

69.  Four  men  hired  a  coach  for  $  13,  to  convey  them  to 
their  respective  homes,  which  were  at  distances  from  the 
place  of  starting  as  follows:  A's  16  miles,  B's  24  miles, 
C's  28  miles,  and  D's  36  miles.     What  ought  each  to  pay  ? 

70.  What  is  a  pile  of  wood  8  feet  long,  7  feet  wide,  and 
5  feet  high  worth,  at  $  4.50  per  cord  ? 

71.  When  bank  stock  sells  at  a  discount  of  1\  per  cent, 
what  amount  of  stock,  at  par  value,  will  %  3700  purchase  ? 

72.  The  pound  sterling  is  worth  $4.8665.  How  much 
U.  S.  coin  would  it  require  to  pay  a  debt  of  £  780  18s.  lid.  ? 

73.  A  merchant  imported  120  tons  of  English  iron,  cost- 
ing 1£  pence  per  pound,  on  which  he  paid  a  duty  of  20  per 
cent.  The  freight  was  5  shillings  sterling  per  ton.  What 
was  the  total  cost  in  U.  S.  currency  ?  (1  ton =2240  pounds. 
£1  =  $4.8665.) 

74.  How  many  rods  of  fence  are  required  to  enclose  a 
square  lot  whose  area  is  5184  square  feet  ? 

75.  Property  worth  $6000  is  insured  for  {  of  its  value, 
at  f  of  one  per  cent.  What  will  be  the  loss,  including  pre- 
mium, in  case  of  total  destruction  by  fire  ? 

76.  How  many  acres  of  land,  in  the  form  of  a  square, 
may  be  enclosed  by  160  rods  of  fence  ? 


Review. 


409 


77.  Find  the  square  root  of  .441  correct  to  two  decimal 
places. 

78.  Reduce  17  lb.  10  oz.  Avoirdupois  weight  to  pounds, 
ounces,  pennyweights,  and  grains,  troy  weight.  (1  pound 
Avoirdupois  =  7000  Troy  grains.) 

79.  Reduce  |4if  to  ^s  lowest  terms. 

80.  Find  the  solid  contents  of  a  cube,  the  area  of  one  face 
of  which  is  256  square  feet. 

81.  A  car  contains  21,643  pounds  of  wheat.  Find  the 
value  of  the  load  at  92  ^  per  bushel  of  60  pounds. 

82.  Find  the  area  of  a  triangle  whose  base  is  22  ft.  8  in., 
and  altitude  19  ft.  9  in. 

83.  The  list  price  of  a  certain  stove  is  $  38,  and  the 
retail  dealer  is  allowed  commercial  discounts  of  20  per  cent, 
5  per  cent,  and  3  per  cent.  What  price  does  he  pay  for  the 
stove  ? 

84.  If  a  ton  of  coal  lasts  a  family  21  days,  what  will  be 
the  cost  of  coal  used  by  it  from  Oct.  17,  1904,  to  April  25, 
1905,  exclusive  of  either  day  named,  at  $  4.50  per  ton  ? 

85.  Find  the  cost  of  a  pile  of  4-foot  wood,  27  feet  long 
and  6  feet  high,  at  $  5.50  per  cord. 

86.  How  many  rods  of  fence  will  be  required  to  enclose 
a  field  in  the  form  of  a  right-angled  triangle,  whose  area  is 
13i  acres  and  whose  base  measures  48  rods  ? 

87.  What  is  the  balance  of  a  bill  of  $  64.50,  after  two 
discounts  have  been  made;  the  first  of  20%  on  the  $  64.50, 
the  other  of  5  %  on  what  then  remained  ? 

88.  There  was  shipped  to  Liverpool  from  New  York  in 
one  week  $  6,870,205  in  specie.  What  amount  of  English 
currency  could  be  bought  with  it  ?     (£  1  =  $  4.8665.) 

89.  What  is  the  freight  on  9860  pounds  iron  at  $  1.75  per 
ton? 


41  o  Chapter  Six. 

90.  What  is  the  value  of  10  lb.  7  oz.  16  pwt.  of  gold  at 
9  -75  a  pennyweight  ? 

91.  The  dividend  is  6171,  the  quotient  17,  the  remainder 
102.     What  is  the  divisor  ? 

92.  Divide  the  L.  C.  M.  of  132  and  156  by  their  G.  C.  D. 

93.  The  product  of  three  numbers  is  .0728  j  one  of  them 
is  1.3,  another  .07.     Find  the  third. 

94.  If  5  men  can  make  38  rd.  5  yd.  of  fence  in  a  day, 
how  much  can  they  build  in  30  days  ? 

95.  The  distance  from  New  York  to  New  Haven  being 
73  mi.  8  rd.,  at  what  rate  does  a  train  run  per  hour  to  cover 
the  distance  in  2  hr.  10  min.  ? 

96.  Eeduce  4  da.  4  hr.  48  min.  to  the  decimal  of  a  week. 

97.  After  4  per  cent  of  a  flock  of  sheep  had  been  killed 
by  dogs,  and  68  had  been  sold  to  a  butcher,  four-sevenths 
of  the  original  flock  were  left.  Required  the  number  of 
sheep  in  the  flock  at  first. 

98.  Six  men  bought  a  ship  worth  $  45,268,  for  which  A 
paid  \  of  the  whole,  B  J,  and  the  others  paid  the  remainder 
equally.     How  much  did  each  of  the  latter  pay  ? 

99.  A  man  agrees  to  dig  a  cellar  30  feet  long,  24  feet 
wide,  and  6  feet  deep.  What  per  cent  of  the  work  has  he 
done  when  he  has  removed  16  cubic  yards  ? 

100.  How  many  boards  16  feet  long,  and  4  inches  wide, 
are  required  to  floor  a  room  48  feet  long,  and  32  feet  wide  ? 

101.  How  much  walking  does  a  man  save  by  crossing 
diagonally  a  field  28  rods  long,  and  21  rods  wide,  instead  of 
going  along  the  end  and  the  side  ? 

102.  In  order  to  have  an  annual  income  of  $2500,  what 
sum  must  be  invested  at  5%  ? 

103.  At  $2  a  rod,  what  is  the  difference  in  the  cost  of 
fencing  a  lot  of  land  20  rods  square,  and  another  lot  contain- 
ing the  same  area  which  is  40  rods  long  ? 


Review.  41 1 

104.  If  a  man  owning  45%  of  a  steamboat  sells  J  of  his 
share  for  $  5860,  what  is  the  value  of  the  whole  boat  ? 

105.  A  farmer  having  6  bu.  8  qt.  of  cranberries  lost  by- 
decay  7  pk.  7  qt.     What  %  had  he  left  ? 

106.  Sold  tea  for  114%  of  its  cost,  and  made  a  profit  of 
7^  a  pound.     Find  selling  price. 

107.  In  §  of  an  acre  of  land  how  many  building  lots,  each 
60  feet  by  121  feet? 

108.  I  bought  a  store  for  a  certain  sum,  and  after  paying 
a  tax  of  2\°/o  on  the  cost  and  \<fo  more  for  insurance  I  sold 
it  for  $  7828,  which  exactly  covered  the  cost,  tax,  and  insur- 
ance.    What  was  the  cost  ? 

109.  Parker  P.  Simmons,  of  Vermont,  sent  to  Nostrand 
Bros,  of  Boston,  to  be  sold  on  commission,  the  following 
goods :  25  tons  of  hay,  2  tons  of  butter,  1500  pounds  of 
maple  sugar,  75  gallons  maple  syrup.  Nostrand  Bros,  sell 
the  hay  at  $  18  a  ton,  the  butter  at  20  tf  a  pound,  the  sugar 
at  7P  a  pound,  the  syrup  at  90^  a  gallon. 

Nostrand  Bros,  charge  2%  commission.  How.  much  da 
they  send  to  Parker  P.  Simmons  ? 

110.  How  much  will  a  granite  block  weigh  which  is 
7  feet  long,  2  ft.  6  in.  wide,  3  ft.  4  in.  high  ?  (12  cubic- 
feet  of  granite  weigh  a  ton.) 

111.  A  coal  dealer  bought  350  tons  of  coal,  weighing- 
2240  pounds  each,  at  $3.50  a  ton.  He  sold  the  coal  at 
$4.25  a  ton,  each  ton  weighing  2000  pounds.  What  was- 
his  profit  ? 

112.  Mrs.  Burns  buys  40  yards  of  carpet  f  of  a  yard 
wide.  She  uses  10%  of  it  for  a  rug,  and  the  remainder  to 
carpet  a  floor.  How  many  square  yards  does  she  use  for 
the  floor  ? 

113.  Mr.  Burns  sold  his  carriage  for  $224,  which  was  f 
of  its  cost.  What  per  cent  would  he  have  gained  if  he  had 
sold  it  for  $210? 


412  Chapter  Six. 

114.  What  is  the  difference  between  four  thousand  nine 
and  seven  hundred  eighty-six  ten-thousandths,  and  four  hun- 
dred thousand  nine  and  seven  hundred  eighty-six  millionths  ? 

115.  Discover  a  fraction  which,  multiplied  by  %,  equals  -J. 

116.  What  %  of  i  of  |  of  |  is  %? 

117.  How  many  inches  in  y1^  of  a  mile  ? 

118.  Bought  a  horse  for  $90,  and  sold  him  for  $95. 
What  per  cent  of  gain?  Bought  another  horse  for  $95, 
and  sold  him  for  $90.     What  per  cent  was  lost  ? 

119.  Bought  land  at  $62.50  per  acre,  and  sold  it  again  at 
$75  per  acre,  thereby  making  $8468.75.  How  many  acres 
were  bought  ? 

120.  Two  ships  sail  from  the  same  port;  one  goes  due 
north  128  miles,  and  the  other  due  east  72  miles.  How  far 
are  the  ships  from  each  other  ?     Illustrate. 

121.  B  and  C,  trading  together,  find  their  stock  to  be 
worth  $3500,  of  which  C  owns  $2100.  They  have  gained 
40%  on  their  first  capital.     What  did  each  put  in  ? 

122.  A  general  wished  to  remove  80,000  pounds  of  provi- 
sions from  a  fortress  in  9  days.  It  was  found  that  in  6  days 
18  men  had  carried  away  but  18  tons.  How  many  men 
would  be  required  to  carry  away  the  remainder  in  3  days  ? 

123.  A  schoolroom  is  40  feet  long,  30  feet  wide,  and  14  feet 
high.  Find  the  difference  between  the  length  of  a  diagonal 
drawn  on  the  floor  and  one  drawn  from  the  floor  to  the  ceiling. 

124.  Find  the  solid  contents  and  the  surface  of  a  sphere 
12  inches  in  diameter. 

125.  The  number  of  copies  in  the  first  edition  of  the 
"  Lady  of  the  Lake  "  was  2050,  and  was  to  the  number  in 
the  second  edition  as  41  to  69.  Find  the  number  in  the 
second  edition. 


Review.  413 

126.  Find  the  proceeds  of  the  following  note : 

$  1050 AV  Chicago,  Feb.  13,  1905. 

Six  months  after  date  I  promise  to  pay  to  the  order  of 
John  G-.  Agar  One  Thousand  Fifty  Dollars,  with  interest  at 
6  per  cent.  Henry  R.  M.  Cook. 

Discounted  at  8  per  cent,  May  13. 

127.  A  can  do  ^  of  a  piece  of  work  in  4  days,  and  B  can 
do  \  of  it  in  5  days.  In  what  time  can  they  do  the  whole 
work  together  ? 

128.  A  square  is  inscribed  in  a  circle  whose  diameter  is 
84  inches.  Find  the  area  of  the  four  segments  of  the  circle 
outside  of  the  square. 

129.  Find  the  difference  between  the  volume  of  a  cylinder 
whose  diameter  and  height  are  12  inches,  and  the  volume  of 
a  sphere  whose  diameter  is  the  same. 

130.  A  certain  cistern  can  be  filled  by  one  pipe  in  10 
hours,  by  another  in  6  hours,  and  can  be  emptied  by  a  third 
in  5  hours.  In  how  many  hours  can  it  be  filled  if  all  three 
pipes  are  opened  at  once  ? 

131.  Two  men  start  from  two  towns  105  miles  apart  and 
walk  toward  each  other.  They  meet  at  the  end  of  15  hours. 
The  first  has  travelled  3  miles  per  hour.  At  what  rate  has 
the  second  travelled  ? 

132.  If  a  cipher  is  added  at  the  right  of  the  decimal, 
what  effect  has  this  on  the  value  of  the  decimal  ?  Explain 
the  reason. 

133.  What  is  the  easiest  method  of  dividing  a  decimal 
by  10? 

134.  If  the  numerator  of  a  common  fraction  is  divided 
by  3,  what  is  the  effect  upon  the  value  of  the  fraction  ? 

135.  If  the  denominator  is  divided  by  3,  what  is  the 
effect  upon  the  value  of  the  fraction? 


414  Chapter  Six. 

136.  What  is  the  effect  on  the  value  of  a  decimal  of 
moving  the  decimal  point  two  places  to  the  right?  Explain 
the  reason. 

137.  What  is  the  exact  interest  on  $400,  from  March  1 
to  December  17,  at  5  per  cent  ?     (365  days  to  the  year.) 

138.  In  what  time  will  a  principal  amount  to  2\  times 
itself,  at  10  per  cent  ? 

139.  A  and  B  in  partnership  have  together  a  capital  of 
$7500,  and  gain  $1200.  A's  share  of  the  gain  is  $250. 
What  is  B's  share  of  the  gain  ?  What  is  B's  share  of  the 
capital  ? 

140.  The  circumference  of  a  circle  is  15.708  feet.  What 
is  the  radius  of  it  ? 

141.  The  radius  of  a  circle  is  42.  What  is  the  circum- 
ference of  it  ? 

142.  Find  the  entire  surface  of  a  cylinder  10  inches  long, 
and  8  inches  in  diameter.  Find  the  number  of  cubic  inches 
in  the  same  cylinder. 

143.  Explain  the  reason  for  multiplying  the  second  and 
third  terms  together  and  dividing  by  the  first  term  in  solv- 
ing an  example  in  simple  proportion. 

144.  Divide  thirty-two  hundred-millionths  by  sixty -four 
ten-thou  sandths. 

145.  A,  B,  and  C  gained  by  speculation  $11,480,  of  which 
A's  share  was  twice  as  much  as  C's,  and  B's  five  times  as 
much  as  C's.     How  much  did  each  gain  ? 

146.  A  pole  was  broken  52  feet  from  the  bottom,  and  it 
fell  so  that  the  top  struck  39  feet  from  the  foot,  while  the 
other  end  of  the  broken  portion  remained  attached.  Re- 
quired the  length  of  the  pole. 

147.  Sold  a  horse  so  that  £  of  the  gain  equalled  ^  of  the 
cost.     What  was  the  gain  per  cent  ? 

148.  In  what  time  will  the  interest  on  £57  Is.  8c?.  amount 
to  £  2  lis.  4£&  at  1\  per  cent  per  annum  ? 


CHAPTER  VII. 

ALGEBEAIO  EQUATIONS. 
ONE  UNKNOWN   QUANTITY. 

516.  A  number  increased  by  12  equals  16. 
This  may  be  written,  x  +  12  =  16. 

The   second  way  is    shorter.      Here  x  stands  for  the 
number. 

517.  PreHminary  Exercises. 

Tell  what  x  may  stand  for,  and  write  in  a  short  way  each 
of  the  following : 

1.  A  number  increased  by  5  equals  7. 

2.  6  is  added  to  a  number.     The  sum  is  9. 

3.  4  subtracted  from  a  number  leaves  1. 

4.  12  diminished  by  a  number  has  8  for  a  remainder. 

5.  A  number  is  subtracted  from  10.    The  remainder  is  3. 

6.  10  is  subtracted  from  a  number.    The  remainder  is  3. 

7.  The  number  of  years  of  John's  age  added  to  3  years 
equals  15  years. 

8.  In  two  years  Mary  will  be  11  years  old. 

9.  5  years  ago  Thomas  was  8  years  old. 

10.  If  William  should  add  5  marbles  to  the  number  he 
now  has,  he  would  have  15  marbles. 

11.  If  Kate  spends  10  cents,  she  will  have  15  cents  left. 

12.  When  paying  for  a  top,  Henry  received  7   cents 
change  from  10  cents. 

415 


41 6  Chapter  Seven. 

13.  A  ball  and  a  bat  together  cost  40  cents.     The  bat 
cost  15  cents. 

14.  A  watch  cost  Mr.  Smith  $  60.     He  bought  the  case 
and  the  works  separately.     The  works  cost  $  20. 

15.  The  weight  of  a  loaded  wagon  is  3200  pounds.     The 
load  weighs  2000  pounds. 


518.   Sight  Exercises. 

If 

x  +  7 

=  9, 

then 

X 

=  2, 

because 

2  +  7 

=  9. 

Find  the  value  of  x : 

1. 

x  +   6  =   9. 

9. 

x  +   5  =  15. 

2. 

x-   4=    1. 

10. 

x+   5=   8. 

3. 

x+   5  =   7. 

11. 

x-    7  =  10. 

4. 

10-   x=   3. 

12. 

x  -  10  =  15. 

5. 

x  -10=    3. 

13. 

x  +  20  =  60. 

6. 

12-   x=    8. 

14. 

x  + 15  =  40. 

7. 

x+   2  =  11. 

15. 

a;  +  2000  =  3200. 

8. 

x+   3  =  15. 

16. 

x+    J=    1. 

COEFFICIENTS. 

519.  (1)  3  x  means  3  times  x. 

(2)  2£  a  means  2  J  times  a. 

(3)  ax  means  a  times  a; . 

In  (1),  3  is  the  coefficient  of  x. 
In  (2),  2£  is  the  coefficient  of  a. 
In  (3),  a  is  the  coefficient  of  x. 

Notice  that  the  coefficient  and  its  letter  are  written  side 
by  side.  Is  there  any  sign  between  them  ?  What  sign  is 
understood  ?    What  is  a  coefficient  ? 


Algebraic  Equations.  417 

520.  "Written  Exercises. 

Write  in  a  short  way,  and  tell  what  x  stands  for. 

1.  8  times  a  number  =  64. 

2.  A  butcher  receives  63  cents  for  a  piece  of  meat  at 
9  cents  a  pound. 

3.  2$  yards  of  muslin  cost  25  cents. 

4.  A  lady  paid  40  cents  for  3  spools  of  black  silk  and 
2  spools  of  white  silk  at  the  same  price  per  spool. 

5.  A  man  worked  by  the  day  10  days  on  my  barn  and 
8  days  on  my  house.     For  all  this  work  he  received  $  36. 

6.  A  man  spent  \  of  his  week's  wages  for  a  pair  of 
boots.     The  boots  cost  him  $  3. 

7.  11  times  a  number  less  2  times  the  number  is  27. 

8.  John's  money  is  in  pennies  and  nickels.     He  has  the 
same  number  of  each.     He  has  42  cents. 

521.  Sight  Exercises. 


If 

10a;- 

■7X: 

=  18, 

then 

Sx 

=  18, 

and           • 

X 

=  6. 

Proof  :  60  -  42  =  18. 

Give  value  of  x  at  sight : 

• 

1.   8a;  =  64. 

8. 

lla;-2a;  =  27. 

2.    9x  =  63. 

9. 

3a;  —  2a;  +  5x  =  54. 

3.   3  a; +  2  a;  =  40. 

10. 

10a; -f  8a; -4a;  =  42. 

4.    2|  a;  =  25. 

11. 

2x  +  4a;  =  52  -16. 

5.    ix  =  S. 

12. 

3a;  +  4a;  =  30  -9. 

6.   10a; -f  8a;  =  36. 

13. 

12a;-5a;  =  25  4-10. 

7.   4  a;  —  3  x  =  72. 

14. 

6a;  +  6a;  =  16 +  8. 

41 8  Chapter  Seven. 

522.  Oral  Exercises. 

If  a  =  2,  and  if  6  =  3,  and  if  c  =  7. 

1.  a  +  b  =  ?  8.  ac  —  &  =  ? 

2.  b  —  a=?  9.  abc=? 

3.  a  +  b  +  c  =  ?  10.  ?  =  9. 

4.  c-a  +  6  =  ?  11.  ?  =  21. 

5.  c-a-&  =  ?  12.  ?  =  23. 

6.  a&  =  ?  13.  ?  =  19. 

7.  ab  +  c  =  ?  14.  ?=4. 

523.  Written  Problems. 

1.  A  horse  and  a  wagon  cost  together  $600.  What  is 
the  price  of  each,  if  the  wagon  costs  twice  as  much  as  the 
horse  ? 

Let  x  =  cost  of  horse ; 

then  2  x  =  cost  of  wagon. 

Cost  of  both  =  2  x  +  x  =  600. 

8aj  =  600. 
x  =  200. 
2  x  =  400. 
Ans.  Cost  of  horse,  $  200  ;  of  wagon,  $  400. 

2.  Divide  100  into  two  parts,  one  of  which  shall  be  four 
times  as  large  as  the  other. 

Let  x  =  one  part ; 

then  4  x  =  the  other. 

x  +  4  x  =  100. 

3.  $  18,000  is  divided  among  three  children,  the  second 
of  whom  receives  twice  as  much  as  the  first,  and  the  third 
of  whom  receives  six  times  as  much  as  the  first.  Kequired 
the  share  of  each.  x  %x  qx 


Algebraic  Equations.  419 

4.  In  a  class  of  54  pupils,  there  are  twice  as  many  boys 
as  girls.     How  many  are  there  of  each  ? 

5.  The  sum  of  two  numbers  is  78.  One  is  five  times  as 
large  as  the  other.     What  are  the  numbers  ? 

6.  156  is  equal  to  seven  times  a  number  added  to  five 
times  the  same  number.     Find  the  number. 

7.  The  difference  between  three  times  a  certain  number 
and  nine  times  the  same  number  is  66.    What  is  the  number  ? 

8.  $  27,000  is  divided  among  three  children,  the  second 
of  whom  receives  twice  as  much  as  the  first,  and  the  third 
of  whom  receives  three  times  as  much  as  the  second.  What 
is  the  share  of  each  ? 

9.  The  sum  of  two  numbers  is  72,  and  the  greater  is  5 
times  the  other.     What  are  the  numbers  ? 

10.  John,  Henry,  and  James  have  54  marbles.  Henry 
has  twice  as  many  as  John,  and  James  has  as  many  as  the 
other  two.     How  many  has  each  ? 

11.  The  sum  of  the  ages  of  mother  and  daughter  is  42 
years.  What  is  the  age  of  each,  if  the  mother's  age  is  six 
times  that  of  her  daughter  ? 

12.  A  man  paid  $96  for  an  equal  number  of  hats  and 
■coats,  paying  $  2  apiece  for  the  former  and  $  10  apiece  for 
the  latter.     How  many  of  each  did  he  buy  ? 

Let  x  =  number  of  each, 

then  2  x  =  cost  of  hats, 

10  x  =  cost  of  coats. 

13.  Divide  41  into  four  parts,  the  first  being  twice  the 
second,  the  second  three  times  the  third,  and  the  third  four 
times  the  fourth.        (Let  x  _  the  fourth>) 

14.  The  sum  of  three  numbers  is  180.  The  first  is  double 
the  second,  and  the  third  is  three  times  as  large  as  the  sum 
of  the  other  two.     What  are  the  numbers  ? 


420  Chapter  Seven. 

15.  Mr.  Smith  paid  81  cents  for  sugar  and  flour,  the 
same  quantity  of  each.  For  the  sugar  he  gave  5  ^  per  pound, 
and  for  the  flour  4/  per  pound.  How  many  pounds  of  each 
did  he  buy  ? 

16.  The  length  of  a  rectangular  field  is  24  rods,  its  breadth 
is  x  rods,  its  area  is  456  square  rods.     Find  the  value  of  x. 

17.  It  takes  340  feet  of  fence  to  enclose  a  square  lot. 
What  are  the  dimensions  of  the  lot  ? 

18.  Mrs.  B.  divides  $  120  between  her  son  and  her  daugh- 
ter. She  gives  the  latter  twice  as  much  as  she  gives  the 
former.     What  is  the  share  of  each  ? 

19.  The  earnings  of  a  man  and  his  son  during  January 
amounted  to  $  175,  both  having  worked  the  same  number  of 
days.  The  father's  wages  were  $  4  per  day,  and  the  son's 
wages  were  $  3  per  day.     How  many  days  did  they  work  ? 

20.  The  sum  of  $240  is  divided  among  four  children,  two 
boys  and  two  girls.  Find  the  share  of  each,  if  each  girl's 
share  is  double  that  of  each  boy. 

21.  A  man  worked  twice  as  many  days  as  his  son.  Their 
combined  earnings  amounted  to  $  165.  Find  the  number  of 
days  each  worked,  if  the  father  earned  $4  per  day  and  the 
son  three-fourths  as  much  per  day. 

22.  A  boy's  bank  contains  78  tf  in  dimes,  nickels,  and 
cents.  There  are  twice  as  many  nickels  as  there  are  dimes, 
and  three  times  as  many  cents  as  there  are  nickels.  How 
many  are  there  of  each  ? 

23.  I  paid  75^  more  for  a  roll  of  15-cent  ribbon  than  I  did 
for  a  roll  of  12-cent  ribbon  of  the  same  length.  How  many 
yards  did  each  roll  contain  ? 

24.  A  rectangular  field  whose  length  is  four  times  its 
breadth  requires  250  rods  of  fence  to  enclose  it.  What  are 
the  dimensions  of  the  field  ?     (Make  diagram.) 


Algebraic  Equations.  421 

25.  A  girl  paid  60  cents  for  a  speller  and  a  reader,  the 
cost  of  the  former  being  one-third  that  of  the  latter.  Find 
the  cost  of  each. 

26.  The  sum  of  two  numbers  is  72,  and  the  smaller  is 
one-fifth  of  the  other.     What  are  the  numbers  ? 

Let  x  =  smaller. 

27.  Mary,  Susan,  and  Jane  have  54  hickory  nuts.  Susan 
has  one-half  as  many  as  Mary,  and  Jane  has  as  many  as  the 
other  two.     How  many  has  each  ? 

Let  x  =  number  Susan  has. 


EQUATIONS. 

524.  An  expression  like  3  x  +  16  =  28  is  an  equation. 
3  x  -f- 16  is  the  first  member  of  the  equation. 

28  is  the  second  member  of  the  equation. 

What  is  the  part  of  the  equation  to  the  left  of  the  equal- 
ity sign  called  ?  What  is  the  second  member  of  an  equa- 
tion? What  sign  is  between  the  two  members?  What 
does  this  sign  show  about  the  value  of  the  two  members  ? 
What  is  an  equation  ? 

525.  Written  Exercises. 
Suppose  a  =  2  and  6  =  3. 

Complete  the  equations : 

1.  ab  +  ?  =  7.  6.  ab  +  b  =  5  +  ?a. 

2.  a  +  6-?  =  4.  7.  17  -  ab  =  5a +  ? 

3.  ?a  +  6  =  ll.  8.  17-a6-?  =  5a. 

4.  ab  =  a  +  b  +  ?  9.  12  +  a6  =  26  +  ?a. 
6.   a&  =  12-?6.  10.  12  +  a6-?a  =  26. 


422  Chapter  Seven. 

CLEARING  OF  FRACTIONS. 

526.  Oral  Exercises. 

1.  One-fifth  of  a  number  is  4.     What  is  the  number  ? 

2.  £  of  a  number  is  8.     What  is  £  of  the  number  ? 

3.  %  of  a  number  is  12.     What  is  the  number  ? 

4.  ^  of  a  number  is  10.     What  is  J  of  the  number  ? 

5.  If  |  of  a  number  is  30,  what  is  the  number  ? 

6.  One-half  a  number  added  to  \  of  the  same  number 
equals  what  fraction  of  the  number  ? 

7.  One-half  a  number  added  to  \  of  the  same  number 
equals  30.     What  is  the  number  ? 

8.  One-third  of  a  number  -f-  one-sixth  of  the  number  = 
what  fraction  of  the  number  ? 

9.  One-third  of  a  number  added  to  \  of  the  number  = 
what  fraction  of  the  number  ? 

10.   \x  +  \ x  =  what  fraction  of  x?     |+|=? 

527.  When  x  =  32,  find  the  value  of  three-fourths  of  x ; 

When  —  (3  x  divided  by  4)  =  24,  what  is  the  value  of 

3z?     0f»? 

Find  the  value  of  y,  when  |  =  12.     Of  2y,  when  ^  =  24. 

4z  ^ 

Given  the  equation  —-  =  20 ;  by  what  whole  number  can 
o 

we  multiply  the  first  member  to  get  rid  of  the  fraction  ?  If 
we  multiply  one  member  of  an  equation  by  any  number^ 
what  must  we  do  to  the  second  member  in  order  to  preserve 
the  equality  ? 

If  equals  are  multiplied  by  equals  the  products  are  equal. 


Algebraic  Equations.  423 

528.  Sight  Exercises. 

Give  values  of  x,  y,  z,  etc. : 

1.  |=4.  ,  g+j-u      9.  ?+|=8. 

9.   |  =  8.  6.   |  +  |=5.  10.   f  +  ^=32. 

3.  |=7.  7.   |  +  |=10.  11.   f  +  |=9. 
4                                  Jo  4      o 

4.  ^  =  21.  8.   |  +  |  =  7.  12.   |  +  ^  =  9. 

13.  |-|=2.  15.   |-?=6. 

14.  §-£  =  8.  16.   2_?=5. 
3     12  2     7 

529.  Written  Exercises. 

Find  the  value  of  the  unknown  quantity  (x). 

Multiplying  by  12,  we  have  6x  +  4x  +  3x  =  312. 

2.  .+1+5-44.  •        . 

Multiply  by  6.    6x  +  3x  +  2x  =  264. 

To  clear  an  equation  of  fractions  multiply  each  term  of  both 
members  by  the  least  common  denominator  of  the  fractions. 

3.  ^  +  ^=35.  6*    t*  +  f*«fl2. 

23  7.   ^  +  ^=102. 

4.  *  +  *=49.  3        4 

3     4  8.   21  x  =115. 

5.  *  +  ^  =  28.  9.   1^-^=48. 
2       3  5        3 


424  Chapter  Seven. 


10. 

a?-  — =  156. 

19. 

75  a;     33  x  _  gl 

40 

100       50 

11. 

3fz  =  116. 

20. 

8  a;     2  a;      ^  oa 
— - —  =  ldb. 

¥=27- 

3        5 

12. 

21. 

2^3^4 

13. 

Ux=x27. 

2 

22. 

a;_^_?  =  37. 

14. 

il^=22. 

2     3 

4 

4  a;     2  a;  .  3  a;      0, 

23. 

—  2< 

15. 

2fa;  =  44. 

5        9        4 

16. 

2  a; -f  —  =  33. 
4 

24. 

0  a;  .  £  x      x      Krt 

17. 

3£a;-2fa;  =  45. 

25. 

aj_^  =  80. 
4 

18. 

a;+f=24. 

26. 

a; -f  2a; +  —  =  24. 

5 

7 

530.   Written  Problems. 

1.  Divide  100  into  two  parts,  one  of  which  shall  be  1J 
times  the  other. 

2.  After  losing  ^  of  his  money,  a  man  has  $714.     How 
many  dollars  had  he  at  first  ? 


(* -|=714) 


3.  A  horse  was  sold  for  $240,  the  seller  thereby  gaining 
one-third  of  what  he  originally  paid  for  it.  How  much  did 
he  pay  for  it  ? 


K) 


4.  One-half  of  a  number  added  to  one-fourth  of  the  same 
number  equals  66f .     What  is  the  number  ? 

5.  The  difference  between  f  of  a  number  and  £  of  the 
same  number  is  15.     Find  the  number. 

6.  One  number  is  £  of  another.    Their  sum  is  55.    What 
are  the  numbers  ? 


Algebraic  Equations.  425 

7.  Find  a  fraction  equivalent  to  J,  the  sum  of  its  numer- 
ator and  its  denominator  being  60. 

(Let  7  x  =  numerator  and  8  x  =  denominator.) 

8.  Find  a  fraction  equivalent  to  j-,  the  difference  between 
its  numerator  and  its  denominator  being  24. 

9.  The  sum  of  two  numbers  is  480,  and  the  quotient  ob- 
tained by  dividing  the  greater  by  the  less  is  7.  What  are 
the  numbers  ? 

10.  Find  two  numbers  whose  difference  is  522  and  whose 
quotient  is  30. 

11.  A  boy  buys  apples  at  2^,  pears  at  3^,  and  oranges  at 
4^,  the  same  number  of  each.  How  many  of  each  does  he 
buy,  if  he  pays  81 4  for  all  ? 

12.  A  girl  bought  70  cents'  worth  of  peaches  and  plums. 
She  paid  3^  each  for  the  peaches  and  2^  each  for  the  plums, 
buying  four  times  as  many  of  the  former  as  of  the  latter. 
How  many  of  each  did  she  buy  ? 

13.  $  1500  is  divided  among  three  persons,  the  second  of 
whom  receives  three  times  as  much  as  the  first,  and  the  third 
three  and  one-half  times  as  much  as  the  first.  Find  the 
share  of  each. 

14.  A  farmer  paid  for  a  cow  three-sevenths  as  much  as 
he  paid  for  a  horse.  How  much  did  he  pay  for  each,  if  the 
latter  cost  $80  more  tl-  n  the  former  ?  j 

15.  Three  times  a  man's  money  increased  by  two-thirds 
of  his  money  is  equal  to  $  1100.    How  much  money  has  he  ? 

16.  After  giving  away  f  of  his  marbles  'and  losing  \  of 
them,  Joseph  has  24  left.     How  many  had  he  at  first  ? 

17.  Bought  a  coat,  a  hat,  and  an  umbrella  for  $15,  pay- 
ing for  the  hat  1^-  times  as  much  as  for  the  umbrella,  and 
for  the  coat  3^  times  as  much  as  for  the  hat.  Find  the  price 
of  each. 


426  Chapter  Seven. 

18.  A  merchant  purchased  two  pieces  of  cloth  for  $240, 
paying  for  one  piece  twice  as  much  per  yard  as  for  the  other. 
The  former  contains  36  yards  and  the  latter  48  yards.  How 
much  does  he  pay  per  yard  for  each  ? 

19.  A  farmer  sold  4  times  as  many  cows  as  horses,  receiv- 
ing for  all  $840,  at  the  rate  of  $40  for  a  cow  and  $120  for 
a  horse.     How  many  of  each  did  he  sell  ? 

20.  The  weight  of  a  team  with  a  loaded  wagon  is  5500 
pounds.  The  wagon  weighs  f  as  much  as  the  load.  The  team 
weighs  twice  as  much  as  the  wagon.  How  many  pounds 
does  the  load  weigh  ? 


53i.   Oral  Exercises. 

Give  values  of  x,  y,  z, 

etc.: 

1.   a +  15  =  21. 

7. 

Sy  +  6  =  15. 

2.   22/  +  15  =  21. 

8. 

7  y  -  13  =  15. 

3.   0-7  =  21. 

9. 

9  y  +  13  =  58. 

4.   4w-7  =  21. 

10. 

3y-10  =  56. 

5.   |  +  3  =  8. 

11, 

5^+1  =  7. 
4 

6.    --3  =  12. 

12. 

i^-i=n. 

532.   If  x  +  15  =  21,  x  =  21  -  what  ? 

When  x-  7  =  21,  x  =  21  +  what  ? 

If  in  the  equation  2  x  -f 15  =  21,  we  take  away  15  from 
the  first  member,  what  must  we  do  to  the  second  member  to 
preserve  the  equality  ? 

If  equals  are  subtracted  from  equals,  the  remainders  are 
equal. 

'  By  transposing  we  mean  bringing  the  unknown  quantities 
(jr,  /,  z,  etc.)  to  one  side  of  the  equation  and  the  known  quan- 
tities to  the  other. 

Note.  —  In  bringing  a  quantity  from  one  side  of  the  equation  to  the 
other,  the  sign  of  the  quantity  is  changed. 


Algebraic  Equations.  427 

533.   Written  Exercises. 

Find  values  of  the  unknown  quantities. 

Note.  —  Clear  of  fractions  when  necessary  ;  then  transpose. 


1.   x  +  37  =  56. 

5. 

x  +  3  x  =  25  +  11. 

2.   4a;-5  =  83. 

6. 

5  x  =  x  +  40. 

3.   3  a; -43  =  98. 

7. 

3  x  -  20  =  x  -  8. 

4.    7x  +  13  =  lll. 

8. 

12  -  3  x  =  45  -  4  x. 

9.   3z-6  =  48  +  z. 

15. 

7rc-5a;  =  20  +  a;+4 

0.   3z  +  6  =  9-2a  +  12. 

16. 

6x  -14  =  16  +  x 

1.   2z-2-16  =  a;  +  10. 

17. 

2z-ll+6a;-60=5a; 

2.   --8  =  24. 

18. 

l+|-6=10- 

3.   ^  +  4-7  =  21. 
6 

19. 

2*-6  =  16  +  |-|. 

14.    |  +  ^  =  10  +  5.  20.    2z  +  ^-|  =  ^  +  27. 

534.   Written  Problems. 

1.  The  sum  of  three  numbers  is  51.  The  second  is  5 
less  than  the  first,  and  the  third  is  10  less  than  the  first. 
What  are  the  numbers  ? 

Let  x  =  first  number, 

x  —  5  =  second  number, 
x  —  10  =  third  number  ; 
x  +  x  -  5  +  x  -  10  =  51. 
Transposing,  x  +  x  +  x  =  51  -f  5  +  10, 

3  x  =  66, 
x  =  22,  first  number, 
a;  —  5  =  17,  second  number, 
X  —  10  =  12,  third  number. 


428  Chapter  Seven. 

2.  Add  45  to  four  times  a  number,  and  you  will  have 
seven  times  that  number.     What  is  the  number  ? 

(7  x  =  45 -f  4  x.) 

3.  Nine  times  a  number  less  27  equals  six  times  the 
number.     Find  the  number. 

4.  Two  boys  have  together  48  marbles.  One  has  18 
more  than  the  other.     How  many  has  each? 

(x,  x+  18.) 

5.  The  length  of  a  rectangular  lot  is  75  feet  more  than 
the  breadth.  The  distance  around  it  is  250  feet.  What  are 
its  dimensions  ? 

6.  A  piece  of  land  containing  86  acres  is  to  be  divided 
into  two  fields,  one  of  which  shall  be  8  acres  larger  than  the 
other.     How  many  acres  in  each  field  ? 

7.  At  a  certain  election  2436  votes  were  cast  for  two 
candidates,  the  successful  one  receiving  318  more  votes  than 
his  opponent.     How  many  votes  did  each  receive  ? 

8.  A  man,  being  asked  his  age,  replied  that  if  he  were 
half  as  old  again  and  7  years  more  he  would  be  100.  What 
was  his  age  ? 

9.  The  sum  of  two  numbers  is  96,  and  their  difference  is 
72.     Find  the  numbers. 

(Let  x  =  less,  x  +  72  =  greater.) 

10.  After  paying  £  and  J  of  my  debts,  I  still  owe  $  45. 
How  much  did  I  owe  originally  ? 

s_?_?  =  45. 
3     4 

11.  Divide  45  into  two  parts,  one  of  which  shall  be  6  less 
than  twice  the  other. 

12.  William  has  $5  more  than  John,  and  three  times 
William's  money  added  to  five  times  John's  would  be  $  103. 
How  many  dollars  has  each  ? 


Algebraic  Equations.  429 

13.  I  bought  3  cows  and  4  horses  for  $635,  paying  $  80 
apiece  less  for  the  cows  than  for  the  horses.  How  many 
dollars  apiece  did  I  pay  for  each  ? 

14.  Mary  has  a  dollar  in  dimes  and  five-cent  pieces. 
She  has  11  more  of  the  latter  than  of  the  former.  Find 
the  number  of  pieces  of  each  denomination. 

15.  Divide  100  into  two  parts  whose  difference  shall 
be  48. 

16.  In  a  class  of  54  pupils,  the  girls  outnumber  the 
boys  by  12.     How  many  are  there  of  each? 

17.  $  18,000  is  divided  among  three  persons,  the  second 
of  whom  receives  9  2400  more  than  the  first,  and  the  third 
of  whom  receives  $2400  more  than  the  second.  Find  the 
share  of  each. 

18.  The  greater  of  two  numbers  is  11  more  than  3  times 
the  less.     Their  difference  is  33.     What  are  the  numbers  ? 

19.  A  boy  spent  a  dollar  for  postal  cards,  2-cent  stamps, 
and  5-cent  stamps.  He  bought  15  more  2-cent  stamps  than 
5-cent  stamps,  and  15  more  postal  cards  than  2-cent  stamps. 
How  many  of  each  did  he  buy  ? 

Let  x  =  number  of  5-cent  stamps, 

then  x  +  15  =  number  of  2-cent  stamps, 

x  -f-  30  =  number  of  postal  cards. 

5  x  =  value  of  5-cent  stamps, 
2  x  +  30  =  value  of  2-cent  stamps, 
x  +  30  =  value  of  postal  cards. 
5£-f-2s  +  30  +  a  +  30  =  100 

20.  A  farmer  has  88  head  of  stock  —  horses,  cows,  and 
sheep.  He  has  17  more  cows  than  horses,  and  the  number 
of  sheep  is  22  greater  than  that  of  the  cows  and  horses 
together.     How  many  are  there  of  each  ? 


430  Chapter  Seven. 

ADDITION  OF  ALGEBRAIC   QUANTITIES. 
535.   Oral  Exercises. 
Add: 


1.   2  fours 

2.    6  hundredths       i 

3.    $4       4. 

3f 

5.    7  a? 

3  fours 

8  hundredths 

$5 

5? 

4a? 

4  fours 

10  hundredths 

$7 

*f 

2x 

5  fours 

12  hundredths 

$3 

9^ 

5x 

?  fours 

?  hundredths 

i? 

U 

~x 

When  no 

coefficient  is  expressed, 

1  is  understood. 

Thus, 

abc  =  1  abc. 

Where  nc 

>  sign  is  expressed,  +  is 

understood 

. 

6.   —  2a 

7.   +  3x     8.   -5xy 

9.     9  abc 

10. 

— 24:xyz 

-  4a 

+  4a;           —  4:xy 

15  abc 

—  5xyz 

-  6a 

+  5a?           —    xy 

6  abc 

-    xyz 

-  la 

+10  a;           —2xy 

abc 

—15xyz 

-19  a 

+  ?  x           -?xy 

?  abc 

—  ?  xyz 

NEGATIVE  QUANTITIES. 

536.  Suppose  three  men  as  follows : 

The  first  man  has  $5  and  owes  nothing. 
The  second  man  has  $5  and  owes  $5. 
The  third  man  has  nothing  and  owes  $5. 
The  first  man  is  worth  $5. 
The  second  man  is  worth  nothing. 

The  third  man  is  worth  5  less  than  nothing.     So  we  may 
say  he  is  worth  —  $  5. 

537.  Quantities  like  —  $5,  —17,  and  —2a  are  called  negative 
quantities. 

What  sign  precedes  a  negative  quantity  ? 

Quantities  with  a  plus  sign  expressed  or  understood  are  called 
positive  quantities. 


Algebraic  Equations.  433 

SUBTRACTION  OF  ALGEBRAIC   QUANTITIES. 

541.  Preliminary  Exercises.  * 

1.  A  man  sold  a  horse  for  $  100  at  a  gain  of  $  25.    Find 
the  cost.     (Cost  =  selling  price  —  gain.) 

$  100  =  selling  price  $  100 

subtract  25  =  gain  or  add  —  25 

remainder  $  75  =  cost  $   75 

2.  A  man  sold  a  horse  for  $  100  at  a  gain  of  —  $  25. 
Find  the  cost. 

$  100  =  selling  price  $  100 

subtract  -  25  =  gain  or  add  -f  25 

$  125  =  cost  $  125 

In  the  first  of  the  above  examples,  subtracting  4-  $  25  is  the  same 
as  adding  —  $  25. 

In  the  second  of  the  above  examples,  subtracting  —  $  25  is  the  same 
as  adding  +  $  25. 

We  changed  the  first  example  from  subtraction  to  addition  by 
changing  the  sign  of  the  subtrahend  from  +  to  — . 

We  changed  the  second  example  from  subtraction  to  addition  by 
changing  the  sign  of  the  subtrahend  from  —  to  +. 

To  subtract  in  algebra,  change  the  sign  of  the  subtrahend 
and  proceed  as  in  addition. 

3.  Add  7  and  -  3.  6.    Add  -  7  and  3. 

4.  From  7  subtract  -  3.         7.    Add  -  7  and  -  3. 

5.  From  —  7  subtract  3.         8.    From  —  3  subtract  —  7. 
9.    Subtracting  —  7  is  the  same  as  adding  what  ? 

10.  Is  a  positive  quantity  increased  or  decreased  by  sub- 
tracting a  negative  quantity  ? 

Note.  —  When  you  become  familiar  with  the  process  of  subtraction 
it  will  not  be  necessary  to  write  the  subtrahend  with  a  changed  sign. 
You  can  conceive  the  sign  changed  and  add. 


43 2  Chapter  Seven. 

We  may  put  down  the  above  statements  thus : 

$     6  $-3 

-3  2 

4  -2 

8  4 

-2  -1 

_I  -2 

$20  $-2 

In  the  first  example  we  add  all  the  positive  quantities,  $  6  -f  $  4 
+  $8  +  $7  =  $25.     Then  we  add  all  the  negative  quantities,  —  $3 

—  $  2  =  —  $  5.     Adding  $  25  and  —  1 5  the  result  is  $  20. 

In  the  second  example  we  add  all  the  positive  quantities,  and  get 
$  6.     The  sum  of  the  negative  quantities  is  —  $  8.    Adding  $  6  and 

—  $  8  the  result  is  —  $  2. 

Can  you  give  the  rule  for  addition  where  the  quantities 
have  different  signs  ?     Which  sign  does  the  sum  take  ? 

540.   Written  Exercises. 
Add: 

1.    —2a     2.        7x     3.    —  5xy  4.    —  9abc  5.   —  24an/z 

—  4:xy  15  abc  5  xyz 

xy  6  abc  xyz 

2xy  —    abc  15  xyz 

6.  3a; +  14,  —7a; +  9,  -23, 4a; -5,  -2a;,  and  3a; +  11. 

3a; +  14 
Write  like  quantities  in  the  same  column.        __  j  x  ,     g 

Find  the  sum  of  the  positive  terms,  also  —23 

the  sum  of  the  negative  terms;  subtract  4#—   5 

the  less  from  the  greater,  and  prefix  the  __2a; 

sign  of  the  greater.  3  a; +  11 

7.  4a  +  3a;,  —2a,  —7a;  — 3a,  —5a;,  —  9a  +  a;. 

8.  -36  + c,  4a +  6 6,  56-9 c,  -3a,  -2a-3&  +  4& 

9.  ix-S,  -a;  +  4,  —  \x  —  3,  7a;  +  16,  -5a;-10. 
10.  4a;  +  23,  -%x  +  2\,  -|a;  +  ll,  -a;  +  5,  9a;-3. 


-2a 

2. 

7x 

-4a 

—  4a; 

—  6a 

-2a; 

la 

5x 

Algebraic  Equations.  433 

SUBTRACTION  OF  ALGEBRAIC   QUANTITIES. 
541.  Preliminary  Exercises.  • 

1 .  A  man  sold  a  horse  for  $  100  at  a  gain  of  $  25.    Find 
the  cost.     (Cost  =  selling  price  —  gain.) 

$  100  =  selling  price  §  100 

subtract  25  =  gain  or  add  —  25 

remainder  $   75  =  cost  $   75 

2.  A  man  sold  a  horse  for  $  100  at  a  gain  of  —  $  25. 
Find  the  cost. 

1 100  =  selling  price  $  100 

subtract  —  25  =  gain  or  add  +  25 

$  125  =  cost  $  125 

In  the  first  of  the  above  examples,  subtracting  +  $  25  is  the  same 
as  adding  —  $  25. 

In  the  second  of  the  above  examples,  subtracting  —  $  25  is  the  same 
as  adding  +  $  25. 

We  changed  the  first  example  from  subtraction  to  addition  by 
changing  the  sign  of  the  subtrahend  from  +  to  — . 

We  changed  the  second  example  from  subtraction  to  addition  by 
changing  the  sign  of  the  subtrahend  from  —  to  + . 

To  subtract  in  algebra,  change  the  sign  of  the  subtrahend 
and  proceed  as  in  addition. 

3.  Add  7  and  -  3.  6.    Add  -  7  and  3. 

4.  From  7  subtract  —3.         7.    Add  —7  and  —3. 

5.  From  —  7  subtract  3.         8.    From  —  3  subtract  —  7. 
9.    Subtracting  —  7  is  the  same  as  adding  what  ? 

10.  Is  a  positive  quantity  increased  or  decreased  by  sub- 
tracting a  negative  quantity  ? 

Note.  —  When  you  become  familiar  with  the  process  of  subtraction 
it  will  not  be  necessary  to  write  the  subtrahend  with  a  changed  sign. 
You  can  conceive  the  sign  changed  and  add. 


434 


Chapter  Seven. 


542.   Sight  Exercises. 

1.  /What  is  the  difference  between  +  52°  and  +33°? 

2.  Between  +  90°  and  -  10°? 
Show  by  a  diagram. 

3.  A  has  $600,  B  owes  $400.     What  are   they  worth 
together  ?  ^+  g  600^  +  (_  $  400)  =  ? 

4.  How  much  better  off  is  A  than  B  ? 

(+$  600)  -(-$400)  =  ? 

5.  From  —  8 a  take  —2a. 

6.  From  —2a  take  8 a. 

7.  From  —  2  a  take  —  8  a. 

8.  From  2  a  take  —  8  a. 

9.  From  3  x +14  take  as +10. 

3a;  +  14 
-   a? -10 

10.  From  5a;— 8  take— 3a;— 9. 

11.  From  a;— 28  take  5  a,*— 37. 

12.  From  7a;+16  take  9a;— 4. 

13.  From  6  x  take  2 x  —  5. 

14.  From  8  x  take  9  x  +  3. 

15.  From  3  a;  +  2  a  — 5  take  a;  — a  — 9. 

16.  From  ly  —  2z  +  b  take  — 8y  +  66  — 2. 

17.  From  c  —  d  +  e  take  c  +  d  —f. 


543.   Written  Exercises. 

1. 

From  8 

a  take  2  a. 

8a 
-   2a 

Ans. 

6a 

2. 

From  2 

a  take  8  a. 

2a 
-   8a 

-4ns. 

-   6a 

3. 

From  - 

-  8  a  take  2  a. 

-  8a 

-  2a 

Ans. 

-10a 

4. 

From  8 

a  take  —  2  a. 

8a 
+   2a 

Ans. 

10  a 

Algebraic  Equations.  435 

REMOVING  PARENTHESES. 

544.  Written  Exercises. 

1.   From  6  x  +  15  y  take  4  x  + 10  y. 

We  may  write  the  above  in  a  shorter  way,  thus  : 

6x  +  15y-(4x  +  10  y). 

The  minus  sign  before  the  parenthesis  shows  that  the 
quantity  within  the  parenthesis  is  to  be  subtracted.  What 
sign  is  before  10  y  ?  What  sign  is  understood  within  the 
parenthesis  before  4  x  ?  In  subtraction,  what  is  done  with 
the  signs  of  the  subtrahend?  If  the  whole  expression  is 
written  without  using  the  parenthesis,  what  must  be  done 
with  the  signs  of  the  quantities  within  the  parenthesis  ? 

a  —  (b  —  c)  may  be  written  a  —  b  +  c.     Why  ? 

a  +  (6  —  c)  may  be  written  a  +  b  —  c.     Why  ? 

When  removing  a  parenthesis  preceded  by  a  minus  sign, 
change  the  signs  of  all  quantities  within  the  parenthesis. 

545.  Written  Exercises. 

Write  the  following  without  parentheses : 

1.  57  +  (33  -  16)  =  74.  4.    (17  -  8)  -  (16  - 14)  =  7. 

2.  92 -(63 +  25)  =4.  5.   75  +  4  x  (15  -  10)  =  95. 

3.  (43 -10) +  (24 -5)  =  52.     6.    75  -4  x  (15-  10)  =  55 

7.  4  x  +  5  y  +  (2  x  —  6  y)  =  6  x  —  y. 

8.  4z  +  5?/—  (2ic  +  6?/)  =  2a;  —  y. 

9.  4:X  —  5y  —  (x  —  6y)  =  Sx-{-y. 

10.  4#  —  5?/  —  (— ic+  6?/)  =  5<c  —  11  y. 

11.  4X+5?/  —  (—  2a  —  6y)=t?. 

12.  -4  0  -5  y  +  (2  0  -6  ?/)  =  ?. 


436  Chapter  Seven. 

546.  Solve  the  following  equations.     Prove  the  correct- 
ness of  your  answers. 

1.  6(2  x  -5)  =  5  x  +  12. 

Note.     6(2  x  —  5)  means  6  times  (2  x  —  6),  or  12  x  —  30. 

2.  7(x  +  2)  =  3x  +  50.  4.   3(16  -  a?)  =  4(13  -  x). 

3.  5(3  + a?) +  16  =  61.  5.   15(x  -  3)  =  2(189  -  16  x) 

6.  38 -(11 -9  a)  =  10  a. 
Removing  the  parenthesis,  we  have 

38  -  11  +  9  x  =  10  x. 

Transposing,  9  x  -  10  x  =  -  88  +  11, 

or,  -  x  =  -  27. 

Bringing  —  x  to  the  right  side  of  the  equation,  and  —  27  to  the  left 
side,  we  have  (+)27  =  (+)x 

In  practice,  however,  when  the  result  is  such  as  the  above,  —  x  = 
—  27,  the  signs  of  both  members  are  changed,  and  the  result  becomes 

x=27. 

7.  2(x-l)-2(2x-19)  =  3(x-3). 

8.  6(2  a? -6) -5*  =  12. 

9.  5  x  -  6(2  x  -  5)  =  -  12. 

10.   11-3a?  +  5x  =  19. 

547.  !^_24-4  =  2. 

Z  o 

Clear  of  fractions  by  multiplying  both  members  of  the  equation  by 
10,  and  observe  which  sign  must  be  changed  to  preserve  the  equality. 
When  x  =  6,  the  above  may  be  written 

3s-6     4s-4_2 
2  6 

Clearing  of  fractions,  16  x  -  30  -  (8  x  -  8)  =  20. 


Algebraic  Equations.  437 

Removing  the  parenthesis, 

15x_30-8z  +  8  =  20. 
Transposing,  15  x  -  8  x  =  20  +  30  -  8, 

or,  7x  =  42, 

x  =  6. 

Note.  —  The  horizontal  line  between  the  numerator  and  the  de- 
nominator of  the  foregoing  fractions  has  the  effect  of  a  parenthesis, 
the  entire  quantity  above  the  line  being  divided  by  the  number  below. 

Hence  when  an  equation  is  cleared  of  fractions,  what  must  be  done 
with  the  signs  of  the  terms  obtained  from  a  fraction  with  a  minus  sign  ? 

l8-^=  (18-6)  -2,  24^4  =  ^of  (24-4). 

1     iof(3x-6),  4x~4=(43-4)-*-5. 


2 
548.   Solve 


11.  ^l  +  ^2  =  8. 

2  3 

12.  ^1_E^2  =  2. 

2  3 

13.  ^zl_^zl_^zl  +  2  =  0. 

2  3  4 

'      2x  —  5  ,  x  —  7     5x  —  3 
"•   — 2-  +  — =  — 6" 

15.   l£^_(«  +  2)  =  **±«_!I±2. 

..    40-5a:     52  +  9* 
16-    — —  =  — — ' 

17.  9|-/|,_|\-f|»+3|* 

18.  2as  =  3  +  2Ja!-(5+|a!)+2|. 

19.  fa;  +  9  =  2*  +  (|x-|a!). 

20-  I+H+I+81-* 


438  Chapter  Seven. 

21.  |a>-120  =  |  +  10. 

22.  a? -20  =  (1  +  15^4 

23.  a  +  |  +  |  =  19. 

24.  9(8a  +  l)-4  =  4(9a;  +  5)+3. 

25.  2  a; +  3  =  — 

549.   Written  Problems. 

1.  A  certain  number  is  multiplied  by  3f;  7  is  subtracted 
from  the  product ;  the  remainder  is  divided  by  16,  giving  a 
quotient  of  3.     What  is  the  number  ? 

2.  Three-eighths  of  what  number  is  60  less  than  the 
number  itself  ? 

3.  Four  persons  are  of  the  same  age.  If  the  first  were 
\  of  his  age  older,  the  second  \  of  his  age  older,  the  third  \ 
of  his  age  older,  and  the  fourth  \  of  his  age  older,  the  sum 
of  their  ages  would  be  99  years.     What  is  the  age  of  each  ? 

4.  A  man  spends  \  of  his  earnings  on  board  and  lodging, 
■§-  on  clothing  and  repairs,  and  \  on  sundries.  At  the  end  of 
the  year  he  has  $  280  left.     What  are  his  yearly  earnings  ? 

3  =  ?  +  |  +  |  +  280. 

2i      o       5 

5.  A  boy  gave  \  of  his  marbles  to  one  companion,  and  \ 
of  them  to  another.  He  then  bought  \  as  many  as  he  origi- 
nally had,  and  had  4  marbles  more  than  he  had  at  first. 
How  many  did  he  have  at  first  ? 

6.  A  father's  age  and  a  son's  age  added  together  amount 
to  138  years.  Twelve  years  ago  the  father  was  twice  as  old 
as  the  son.     How  old  is  each  now  ? 

Let  x  =  son's  age  12  years  ago.    2  x  =  father's  age  then. 


Algebraic  Equations.  439 

7.  John  has  80  cents,  and  William  has  60  cents.  How 
many  cents  should  William  give  John  so  that  the  latter 
might  have  2\  times  as  much  money  as  the  former  ? 

After  William  gives  John  x  cents,  the  former  has  (60  —  x)  cents, 
and  the  latter  has  (80  -f  x)  cents. 

8.  In  how  many  years  will  a  man,  now  25,  be  double  the 
age  of  his  11-year-old  brother  ? 

Let  x  =  number  of  years.     25  +  x  and  11  +  x  =  ages  after  x  years. 

9.  A  man  has  a  cask  of  60  gallons'  capacity.  He  draws 
off  one-fourth  of  its  contents,  and  then  fills  it.  If  it  takes 
24  gallons  to  fill  it,  how  many  gallons  did  the  cask  originally 
contain  ? 

10.  A  number  is  divided  by  3,  and  40  is  subtracted  from 
the  quotient,  leaving  a  remainder  of  104.  What  is  the 
number  ? 

11.  The  difference  between  two  numbers  is  430.  When 
the  greater  is  divided  by  the  less,  the  quotient  is  4,  and  the 
remainder  is  76.     What  are  the  numbers  ? 

Let  x  =  less.    EH*Lr  =  4  +  It 
less  less 

12.  A  person  pays  $103  with  29  $2  and  $5  bills.  How 
many  are  there  of  each  denomination  ? 

13.  A  father  is  30  years  older  than  his  daughter.  In 
4  years,  his  age  will  be  four  times  her  age.  What  are  their 
present  ages  ? 

x  and  x  +  30  =  present  ages,     x  +  4  and  x  +  34  =  ages  4  years  later. 

14.  The  product  of  two  numbers  is  180.  If  the  smaller 
number  be  increased  by  3,  the  product  of  the  two  numbers 
will  be  225.     What  are  the  numbers  ? 

smaller  =  x ;  —  =  greater. 

x 

15.  A  man's  wages  are  $  1  per  day  more  than  his  son's. 
For  33  days'  work,  the  father  receives  $  12  more  than  the 
son  earns  in  40  days.     Find  the  wages  of  each. 


440  Chapter  Seven. 

TWO  UNKNOWN  QUANTITIES. 

550.  Preliminary  Problems. 

1.  I  paid  a  dollar  for  two  25^  balls  and  five  bats.  How 
much  did  I  pay  apiece  for  the  latter  ? 

2.  When  three  times  one  number  is  added  to  five  times 
another,  the  sum  is  84.  If  the  second  number  is  12,  what 
is  the  first  number  ? 

3.  A  girl  paid  75^  for  £  pound  of  tea  and  2\  pounds  of 
coffee.  The  coffee  cost  20  ^  per  pound.  What  was  the  price 
of  the  tea  per  pound  ? 

4.  A  man  sold  pigs  at  $5  each  and  lambs  at  $8  each, 
receiving  $42.  He  sold  4  lambs.  How  many  pigs  did  he 
sell? 

5 .  Four  times  a  father's  age  added  to  twice  his  daughter's 
age  amounts  to  180  years.  The  girl  is  10  years  old.  What 
is  the  father's  age  ? 

6.  Eight  peaches  and  seven  pears  cost  44^.  The  peaches 
cost  2^  each.     What  is  the  cost  of  a  pear? 

7.  Two  pieces  of  cloth  and  eleven  pieces  of  silk  contain 
152  yards.  There  are  10  yards  in  each  piece  of  cloth. 
How  many  yards  in  each  piece  of  silk  ? 

8.  Two-thirds  of  a  yard  of  linen  and  three-fourths  of  a 
yard  of  lace  cost  40^.  The  price  of  the  lace  is  32^  a  yard. 
Find  the  price  of  the  linen. 

9.  Three  and  one-half  times  one  number  added  to  four 
and  one-third  times  a  second  number  equals  60.  The  second 
number  is  9.     What  is  the  first  number  ? 

551.  Written  Exercises. 

Find  the  value  of  the  unknown  quantity : 

1.  8  x  +    ly  =  44.     When  a  =  2,  find  the  value  of  y, 

2.  3 y  +    5  z  m  34.     Find  the  value  of  z ;  y  =  3. 

3.  2a  +  ll3  =  152.      a;  =  10;2  =  ?. 

4.  14a  +   7y  =  98.       x  =  3%-,y=?. 


Algebraic  Equations.  441 

5.  !'*+    f*x=40.  2  =  32. 

6.  9x-  25y  =  8.  x  =  12. 

7.  3Jy  +  4i*==eO.  z  =  9. 

8.  16a>-19z  =  49.  z  =  5. 

9.  7y-   3*  =  18.  2/  =  6f 
10.   32x  +  50?/ =2600.  2/  =20. 

552.  Written  Problems. 

1.  The  cost  of  3  apples  and  2  peaches  is  7  cents.  The 
cost  of  2  apples  and  2  peaches  is  6  cents. 

Subtracting  the  second  lot  of  fruit  from  the  first  lot  we  have  1  apple. 
Subtracting  the  price  of  the  second  lot  from  the  price  of  the  first 
lot  we  have  1  cent.     1  apple  costs  1  cent. 
If  equals  are  subtracted  from  equals,  the  remainders  are  equal. 

2.  A  boy  gave  2S$  for  3  lemons  and  8  oranges,  another 
boy  paid  11$  for  3  lemons  and  4  oranges.  How  much  did 
the  lemons  cost  apiece  ? 

x  =  cost  of  lemons,  3  x  +  8  y  =  25  (1) 

y  =  cost  of  oranges,  3  x  +  4  y  =  17  (2) 

Subtracting  (2)  from  (1),  4y=    8 

The  oranges  cost  2f  each,  y  =    2. 

How  much  apiece  was  paid  for  the  lemons  ? 

3.  If  3  coats  and  14  vests  cost  $  78,  and  2  coats  and  14 
vests,  at  the  same  rate,  cost  $66}  how  much  does  1  coat 
cost  ?     What  is  the  price  of  a  vest  ? 

4.  Given  4z  +  ly  =  53,  (1) 

2x  + 32/ =  25,  '  (2) 

to  find  the  value  of  y. 

First  multiply  (2)  by  2,  making  it  4  re  +  6  x  =  50.     Why  ? 

5.  What  is  the  value  of  x  in  equation  (1)  in  above  ex- 
ample, when  the  value  found  for  y  is  substituted  therein  ? 
Substitute  the  same  value  for  y  in  equation  (2)  and  find  the 
value  of  x. 


442  Chapter  Seven. 

553.   Written  Exercises. 

Find  the  values  of  x  and  y  in  the  following  equations : 

1.  x+    2/ =  15,  3.    2x  +  3y  =  18, 
2x  +  3  y  =  38.  4#  +  3y  ==  24. 

2.  2z  +  2y  =  30,  4.   2a;  +  3?/  =  40, 

x  +  32/  =  27.  3z  +  2?/  =  35. 

Multiply  first  equation  by  3,       6  x  +  9  y  =  120. 
Multiply  second  equation  by  2,  6  a;  +  4  y  =    70. 

5.    7  a?  +  5  ?/  =  82,  6.    5  a -f- 9  y  =  14, 

2  *  +  3  ?/  =  36.  9*4  5  ?/ =14. 

7.   3a>  +  53/  =  17,  8.   2x-3y  =  l$, 

8  8.  +  2  2/  =  17.  3z  +  52/  =  65. 

Given  ;ft(  _         .        *  '  \     To  find  values  of  x  and  y. 
(2)  7  #  —  4  2/  =  22.  J  * 

Multiply  (1)  by  7,  7  x  +  21  y  =  322 

(2)     Ix-   4y  =   22    Subtract. 
25  y  =  300 
y=    12 

Substituting  this  value  of  y  in  (1),  we  have 
x  +  36  =  46, 

a;  =  46  -  36  =  10.      Arts,  x  =  10,  y  =  12. 

9.   a;  -J-  y  =  18,    Add  or  subtract. 
a-y-  4. 

10.  4  a;  +  3  #  =  17,  (1)     Multiply  (2)  by  2  and  subtract. 
2a-     y=    I-  (2) 

11.  3  a;  +  4  y  =  48,     Add. 

a?  —  4  y  =    0. 

12.  3  a;  +  5  y  =  13,  (1)     Multiply  (1)  by  7  and  (2)  by  3. 
7x  +  3y  =  13.  (2)     Subtract. 


Algebraic  Equations.  443 

13.  4  x  +  5y  =  32,     Add. 
6a;  —  5y  =  —  2. 

14.  3  4  +  4  2/  =  3,  (1)     Multiply  (2)  by  2.    Add. 
12  x  -2  ?/  =  3.  (2) 

15.  5  a;  =  6  2/  +  5,     Transpose. 
3  x  =  5  y  —  4. 


16.   3z  +  5y+   8  =  0,              17 

y  —  2x=8x  —  l, 

2  as  -    2/  -  12  =  0. 

2y  —  4=x  =  y  +  x-+  9. 

18.   5x+   7y  =   55, 

(1) 

9  x  + 18  y  =  126. 

(2) 

Divide  (2)  by  9  getting  x  +  2  y  =  14. 

C3) 

Then  multiply  (3)  by  5. 

19.      S  +  ?J?  =  17.     Clear  of  fractions. 
4      3 

5a;     5j/  =  20 

4        8 

20.    £a?-f£y  =  42,                     24. 

4^a;  +  3|2/  =  67, 

i*-Hy=F  17* 

7^a;-5i2/  =  12. 

21.   23x-7y=   3a;  +  51,       25. 

3  (a>  +  7)  =9(2/  -9), 

ll2/  =  15a;  +  2. 

4(3  a>  -8)  =  17  y  -155. 

22.    x  -\-y  =  100,000,                  26. 

2(*  - 11)  -2(2/-9)=  6, 

fo+S=^°- 

a?  +  9     32 
y-S     15 

23-  f^I=5> 

»-4     y-l_6 
3            4 

7a;-6_2 
5y  +  3 

a;_4     2/-l_i 
3            4 

28.   2a>4-5*/  +  3  =  6 
3a;-42/-2       ' 

4a;-7?/4-5_^ 

a;-22/  +  2 

444  Chapter  Seven. 

554.   Written  Problems. 

1.  The  sum  of  two  numbers  is  37.  Twice  the  first  added 
to  three  times  the  second  is  96.     What  are  the  numbers  ? 

Let  x  =  first  number ;  y  =  second  number. 

2.  The  difference  between  two  numbers  is  28.  Five 
times  the  first  less  twice  the  second  is  197.  What  are  the 
numbers  ?  x-y  =  28;  hx-2y-  197. 

3.  The  product  of  the  first  of  two  numbers  by  5,  added  to 
the  product  of  the  second  by  3,  gives  37.  The  product  of 
the  first  by  6,  diminished  by  five  times  the  second,  equals 
10.     Find  the  numbers. 

4.  Divide  65  into  two  parts  whose  difference  shall  be  19. 
Let  x  and  y  =  parts.     Solve  also  by  one  unknown  quantity. 

5.  A  person  pays  $103  with  32  bills,  some  of  them  $2 
bills,  the  others  $  5  bills.     How  many  of  each  does  he  use  ? 

6.  For  25  head  of  pigs  and  sheep,  a  farmer  received 
$  145.  How  many  of  each  did  he  sell,  if  he  sold  the  former 
at  $7  each,  the  latter  at  $5  each  ? 

7.  10  oranges  and  4  peaches  cost  38^;  6  oranges  and  7 
peaches  cost  32  ^.     Find  the  cost  of  an  orange.     Of  a  peach. 

8.  5  pounds  of  tea  and  3  pounds  of  coffee  cost  $3.75; 
8  pounds  of  tea  and  1  pound  of  coffee  cost  $  5.05.  What  is 
each  worth  per  pound  ? 

9.  A  farmer  buys  a  certain  number  of  horses  at  $125 
each  and  a  certain  number  of  cows  at  $  40  each.  They  cost 
together  $  740.  If  he  had  bought  half  as  many  horses  and 
twice  as  many  cows  they  would  have  cost  $730.  How  many 
of  each  did  he  buy  ? 

10.  A  man  paid  75^  for  2  pounds  of  raisins  and  3 
pounds  of  cheese.  5  pounds  of  raisins  and  2  pounds  of 
cheese  at  the  same  prices  would  have  cost  94^.  What  did 
each  cost  per  pound  ? 


Algebraic  Equations.  445 

11.  The  sum  of  two  numbers  is  19.  The  sum  of  the 
second  number  and  ten  times  the  first,  minus  the  sum  of 
the  first  and  ten  times  the  second,  equals  45.  What  are  the 
numbers  ? 

12.  Reduce  ^  to  an  equivalent  fraction,  the  sum  of  whose 
numerator  and  denominator  shall  be  126. 

x  =  numerator  ;  y  =  denominator. 

*  =  ±.;x  +  y  =  126. 

y    13 

13.  What  fraction  equivalent  to  -^  has  147  for  the  dif- 
ference between  its  numerator  and  denominator  ? 

x  -  y  =  -  147.     Why  ? 

14.  10  pounds  of  coffee  at  30^  per  pound  are  mixed  with 
x  pounds  of  coffee  at  25^  per  pound.  What  is  x  equal  to, 
when  the  mixture  is  worth  26^  per  pound  ? 

25x  +  (10  x  30)  =  26  (10  +  x). 

15.  A  grocer  mixes  green  tea  costing  60^  per  pound  with 
black  tea  costing  40^  per  pound.  He  uses  100  pounds  in 
all,  and  the  mixed  tea  costs  him  48  ^  per  pound.  How  many 
pounds  of  each  does  he  use  ? 

Let  x  =  number  of  pounds  of  black  tea ;  y  =  number  of  green. 
Then  x  +  y  —  number  of  pounds  of  mixed  tea. 

x  +  y  =  100  ;  40  x  +  60  y  =  48  (x  +  y). 

THREE  UNKNOWN  QUANTITIES. 
555.    1.     Given  the  following : 

3x  +  2y-    2  =  12,  (a) 
5  x  _  4  y  +  3  z  =  16,  (6) 
2  x  +  3  y  +  2  z  =  35,  (c) 
to  find  the  values  of  x,  y,  and  z. 


446  Chapter  Seven. 

(a)  multiplied  by  5,  15  x  +  10  y  -    5  *  =  60 

(6)  multiplied  by  3,  15  x  -  12  y  +    9  z  -  48 

Subtract,  22  y  -  14  z  =  12  (d) 

an  equation  containing  only  two  unknown  quantities. 

(6)  multiplied  by  2,  10  x  -    8  y  +    6  z  =       32 

(c)  multiplied  by  5,  10  a  +  15  y  +  10  z  =      175 

Subtract,  -  23  y  -   4  3  =  -  143  («) 

an  equation  containing  only  two  unknown  quantities. 

Compare  the  two  equations  (d)  and  (e),  which  contain  the  same 
two  unknown  quantities. 

(d)  multiplied  by  2,  44  y  -  28  z  —         24 

(e)  multiplied  by  7,  —  161  y  -  28  z  =  -  1001 

Subtract,  205  y  m     1025 

y  m  5 

Substituting  this  value  of  y  in  (<Z),  we  have 

110  -Uz  =  12,  -  14  z  =  -  98,  z  =  7. 

Substituting  values  of  y  and  z  in  (a),  we  have 

3  x  +  10  -  7  =  12,  3  x  =  9,  a;  =  3. 

^ns.  a;  =  3, 

y  =  5, 

3  =  7. 

2.  Find  the  values  of  the  unknown  quantities  in  the 
following  equations: 

x-Sy  +  2z=  3,  (a) 

2x+    y  +  3z  =  22,  (6) 

5x  +  2y  +  7z  =  5l.  (c) 

Multiply  (a)  by  2,  and  subtract  from  (6).  Multiply  (a)  by  5,  and 
subtract  from  (c).  This  gives  two  equations,  each  of  which  contains 
two  unknown  quantities. 

Compare  these  two  resulting  equations,  and  eliminate  y. 


Algebraic  Equations.  447 

3.  5x-2y+    z  =   10,  (a) 

3x  +  8y-5z  =  120,  (b) 

7x-Sy-2z  =     8.  (c) 

Eliminate  z  by  comparing  (a)  and  (6),  multiplying  the  former  by  5 
Compare  (a)  and  (c),  multiplying  the  former  by  2. 

4.  13ic-   ±y  +  15z  =     317, 

7z  +   2y-   3^=       89, 
21a-17#  +  9z  =  -104. 

5.  -  8x  +      y  —  12z  =  -259, 

7x-   ±y  +  25z=     418, 
13<c  +  2y-4lz  =  -500. 

"e.  §+=±*-H 

x  +  y     x-y.„ 
2  6 

7.  3a;~5y     g==2a?  +  y 

2  5 

8     s  — 2y_a?  .  y 
4  2"V 

8.  2  +  5-^^  =  4^3a;, 

12  +  5a?-6y==2        3^-2^ 
6  4 

2x-\-y     9a;  — 7^3 y  4-9     4a?  +  5y 
2  8      ~      4  16 


448  Chapter  Seven. 

556.   Written  Problems. 

1.  A  man  placed  §  of  his  capital  at  5%  and  the  other 
third  at  6%.  At  the  end  of  a  year,  capital  and  interest 
amounted  to  $  31,600.     What  was  his  capital  ? 

—  x and  -  x  —  =  interest. 

3       100  3     100 

2.  A  has  18  chestnuts  more  than  B.  If  each  finds  4 
more,  A  will  have  four  times  as  many  as  B.  How  many 
chestnuts  has  each? 

3.  Two  mechanics  earn  together  $  8  per  day.  One 
works  23  days  and  tke  other  17  days,  for  which  they  receive 
together  $  166.     What  does  each  earn  per  day  ? 

4.  The  sum  of  the  first  and  the  second  of  three  numbers 
is  55,  of  the  first  and  the  third  62,  of  the  second  and  the 
third  83.     What  are  the  numbers  ? 

Suggestion.  — Add  together  the  three  equations. 

5.  The  sum  of  two  numbers  is  53.  Four  times  the  first 
is  20  more  than  twice  the  second.     Find  the  numbers. 

6.  A  certain  sum  of  money  is  divided  among  four  per- 
sons. The  first  takes  £  of  it,  the  second  takes  f  of  the 
remainder,  the  third  takes  £  of  what  then  remains,  the 
fourth  receives  the  balance,  $24.  What  is  the  share  of 
each  of  the  other  three? 

7.  A  merchant  sold  a  lot  of  goods  for  $510,  thereby 
losing  ^  of  their  cost.     What  did  the  goods  cost  ? 

8.  A  man  collected  a  bill  for  a  physician  and  deducted 
^  of  the  amount  for  his  services.  If  he  gave  the  physician 
$  147,  what  was  the  amount  collected  ? 

9.  Divide  130J  acres  of  land  among  three  persons,  giving 
the  first  27£  acres  more  than  the  second,  and  the  second  13f 
acres  more  than  the  third. 

10.  A  merchant  has  sold  ^  of  a  piece  of  cloth,  and  has 
remaining  16  yards  more  than  £  of  the  piece.  How  many 
yards  did  the  piece  contain  originally  ? 


Algebraic  Equations.  449 

11.  A  servant  is  engaged  for  a  year  for  $280  and  a  suit 
of  clothes  5  he  leaves  at  the  end  of  six  months,  and  receives 
$  130  and  the  suit.     What  is  the  value  of  the  clothes  ? 

Yearly  wages  =  280  +  x.    Wages  for  six  months  =  140  +  ^  =  130  +  x. 


EXPONENTS. 

557.  (1)   x2  means  x  times  x,  or  xx. 

(2)  y3  means  y  times  y  times  y,  or  yyy. 

(3)  a4  means  aaaa. 

In  (1)  2  is  an  exponent. 

In  (2)  3  is  an  exponent. 

In  (3)  4  is  an  exponent. 

Notice  that  the  exponent  is  written  above  the  quantity  to 
which  it  belongs. 

On  which  side  of  the  quantity  is  it  written  ? 

How  many  times  is  the  quantity  used  as  a  factor  if  the 
exponent  is  2  ?  If  the  exponent  is  3  ?  If  the  exponent 
is  5? 

Tell  two  facts  concerning  the  location  of  the  exponent. 
Tell  one  fact  concerning  the  meaning  of  the  exponent. 

What  is  an  exponent  ? 

Note.  —  When  no  exponent  is  written,  1  is  understood. 

558.  Oral  Exercises. 

If  a  =  2,  and  if  b  =  3,  and  if  c  =  5, 

1.  a2  =  ?  7.  a3  +  a2-fa  =  ? 

2.  a2  +  a  =  ?  8.  a3  +  62  +  c  =  ? 

3.  a2  +  b  +  c  =  ?  9.  a4  +  63  +  c2  =  ? 

4.  a2  +  62=?  10.  a2  +  ?c  =  19. 

5.  62-a2  =  ?  11.  a3  +  &3  =  a2-f&2+c2~? 

6.  a2  +  2ab  +  b2=?  12.  b2  +  a4  =  c\ 


45°  Chapter  Seven. 

559.  3  62  means  3  6  times  6  and  not  3  6  times  36.  The 
exponent 2  belongs  to  the  letter  6  and  not  to  the  expression 
3  6. 

ab3  means  a666  and  not  ab  ab  ab. 

In  5  ab3  to  what  part  of  the  expression  does  the  exponent 
belong  ? 

If  we  use  a  parenthesis,  the  effect  is  different.  (36)2 
means  3  6  times  3  6.     (ab)3  means  ab  ab  ab. 

What  does  (2  a)4  mean  ? 

560.  Oral  Exercises. 

If  a  =  2,  and  if  6  =  3,  and  if  c  =  5, 

1.  5a2=?  7.  7ab2  =  ? 

2.  (5a)2  =  ?  8.  7a26  =  ? 

3.  (26)2-262=?  9.  a6c2=? 

4.  4  62-c2  =  ?  10.  a262c  =  ? 

5.  (3a)3-3a3=?  11.  c2 -a2  =  ? (c-a). 

6.  a6c2  =  ?  12.  4  62-2ac  =  a?. 

MULTIPLICATION. 

561 .  What  coefficient  is  understood  when  none  is  written  ? 
What  exponent  is  understood  when  none  is  written  ? 

2  a  times  3     =  6  a. 

2  a  times  3  a  =  6  a2. 

3  a  times  4  a  =  12  a2. 
5  a2  times  2  a  =  10  a8. 
5  a  times  2  6   =  10  ab. 

Multiply  together  the  numerical  coefficients  and  affix  all  the 
letters,  giving  each  letter  the  sum  of  the  exponents  of  that  letter 
in  both  factors. 


Algebraic  Equations.  4ji 

562.  Oral  Exercises. 
Multiply ; 

1.  3a  by  5a.  .7.  5a26by3c. 

2.  2ab  by  3a.  8.  5ab  by  3ac. 

3.  2a6by36.  9.  Sab  by  3  a&V. 

4.  2  a26  by  3  a.  10.  ab2<?  by  a6c. 

5.  2a62by3a.  11.  £a&by6a2. 

6.  2a262by36.  12.  10a26byja. 

SICNS   IN  MULTIPLICATION. 

563.  Preliminary  Exercises. 

1.  -f  a  times  +  b  =  -h  ab. 

Here  the  two  factors,  +  a  and  +  b,  have  like  signs  (both 
are  +).     The  product  has  the  -}-  sign. 

2.  —  a  times  —  b  =  -f  a&. 

Here  the  two  factors  have  like  signs  (both  are  — ).  The 
product  has  the  +  sign. 

3.  —  a  times  -f-  b  =  —  ab. 

Here  the  two  factors  have  unlike  signs  (one  +  and  one  — ). 
The  product  has  the  —  sign. 

4.  -f  a  times  —  b  =  —  ab. 

Here  the  two  factors  have  unlike  signs.  The  product  has 
the  —  sign. 

From  the  above  we  may  see  the  law  for  signs  in  multipli- 
cation. 


Like  signs  give  +• 

Unlike  signs  give  — . 

Multiply : 

5.    —3  a  by  5  a. 

9. 

—  5  a2b  by  —  3  c. 

6.    2  a  by  4  a. 

10. 

-  5  a2b  by  3c. 

7.    —  2  a  by  —  4  a. 

11. 

5a26by  -3c. 

8.   2a62by  -3a. 

12. 

-5a2b  by  -1. 

45 2  Chapter  Seven. 

564.  Written  Exercises. 
Multiply : 

1.  a2  +  3aby2a.  *                               Ans.  2a3  +  6a2. 

2.  4a-6by2a&.  Ans.  8  a26  -  2  ab\ 

3.  5  a  +  3  b  by  4  a.  8.  a2-62bya. 

4.  a«  —  c»  by  —  3  a.  9.  a2  —  62  by  —  a. 

5.  a2  +  a  + 1  by  a.  10.  aW  —  a  by  a&. 

6.  a2  +  2a  +  lby -2a.        11.  a2  +  2a&  +  b2  by  2a&. 

7.  a2-M  +  lbya&.  12.  a2  -2a&  +  62  by  -2ab. 

565.  Written  Exercises. 

1.   Multiply  x  4-  2  by  x  +  3. 

x  +  2 

3  +  3 


Multiplying  x  -f  2  by  x,  x2  4-  2  x 

Multiplying  x  +  2  by  3,  3a;4-6 

Adding  the  two  parts  of  the  product,        x2  +  5  x  4  6 

Multiply  each  term  of  the  multiplicand  by  each  term  of  the 
multiplier  and  combine  the  products. 

2.    Multiply  x  -f-  3  by  x  —  4. 


x  +  3 

x-4 

x2  +  3  a 

• 

-4a 

;-12 

x2-x- 

-12 

Multiply : 

3.   x  4-  3  by  x  +  4. 

7.   2  a;  -8  by  a?  +  9. 

4.   a;  +  5  by  a?  —  2. 

8.   3z  +  l  by  x  +7. 

5.   x  +  8bya;  — 9. 

9.   2a;  +  lby2a:  +  l 

6.   2  x  +  5  by  a;  4-  2.  10.   a;  —5  by  a;  4-  4. 


Algebraic  Equations. 


453 


566.   Written  Exercises. 

Find  products : 

Note  :  (x  —  3)  (x  +  9)  means  x  —  3  multiplied  by  x  +  9. 


1. 

(x  -3)(<c  +  9). 

13. 

(a; -5)  (a; -9). 

2. 

(x-6)  (x  +  7). 

14. 

(a;  -f  5)  (a;  +  5). 

3. 

(a;  -  5)  (a  +  5). 

15. 

(x  -3)  (x  +  8). 

4. 

(x  +  5)(x-5). 

16. 

0  +  7)  (a; -6). 

5. 

(2x-6)(x  +  l). 

17. 

(a._4)(a;-7). 

6. 

(x-6)(2x  +  l). 

18. 

(2a; -4)  (3a; -6). 

7. 

(2  s -6)  (3* +  3). 

19. 

(2x  +  6)  (3a;  -7). 

8. 

(3a? +  6)  (2  a? -3). 

20. 

(2a; +  7)  (3  a; +  3). 

9. 

(2  a;  +  3)  (2  a; -3). 

21. 

(2  a; -3)  (3  a; -2). 

10. 

(a; -5)  (a; -4). 

22. 

(2  a; -3)  (2  a; +  3). 

11. 

(x_  7)  (a; -9). 

23. 

(2  a; +  9)  (4a; -6). 

12. 

(a- 7)  (a -7). 

24. 

(3a;-4)(3a;  +  4) 

567 

.   Written  Exercises. 

1 .   Multiply  x  +  y  by  x  +  y. 

s  +  y 
x  +  y 

x*  +  xy 

xy  +  y* 

x2  +  2xy  +  y* 

Multiply : 

2.  a  +  6  by  a  +  6. 

3.  a  — i/ by  a  — y. 

6.  a;  +  2/  by  x  —  y. 

7.  a  +  lby  a-1. 

8.  m  +  n  by  m  —  w. 

11.  a2  +  a+l  by  a-1. 

12.  a2  — a  +  1  by  a  +  1. 


4.  a  +  3  x  by  a  +  3  x. 

5.  2a +  x  by  a  +  2x. 

Ans.   aP  —  y2. 
9.    a2  +  6  by  a2  +  6. 
10.    a2  +  62bya-6. 

-4ws.   a3  —  1. 
13.   ar*  +  a;+l  by  a;-l. 


454  Chapter  Seven. 

TERMS. 

568.  Preliminary  Exercises. 

1.  2abc.  3.   a2  +  2a&  +  6*. 

2.  2  +  a  +  6  +  c.  4.   lab2  —  !. 

The  first  of  the  above  expressions  has  one  term 

The  second  has  4  terms. 

The  third  has  3  terms. 

The  fourth  has  2  terms. 

An  expression  containing  one  term  is  called  a  monomial ; 
one  containing  two  terms,  a  binomial ;  one  containing  three 
terms,  a  trinomial;  one  containing  four  or  more,  a  poly- 
nomial. 

How  many  terms  has  each  of  the  following  ? 
6.   2  acx2y.  8.   lla2  +  a. 

6.  a  4- 1.  9.   a2  4-  a  +  abc. 

7.  a2  +  a  +  11.  10.    a2  +  a  +  b  +  c. 

LIKE  TERMS. 

569.  Like  terms  are  those  containing  the  same  letters  and 
the  same  exponents  for  each  letter. 

570.  Oral  Exercises. 

'  Which  of  the  following  expressions  contain  like  terms  ? 

1.  a  and  b.  6.    2  a2  and  2  a3. 

2.  2  a  and  a.  7.   2a26  and  3a?b. 

3.  a2  and  a.  8.   2a3b  and  3a26. 

4.  a2  and  2  a.  9.    xy2  and  x*y. 

5.  2  a2  and  3  a2.  10.   barfy  and  ax*y. 


Algebraic  Equations.  455 

COMBINING  LIKE  TERMS. 

571.  Two  or  more  like  terms  may  be  combined  into  a  single 
term.     Unlike  terms  cannot  be  so  combined. 

572.  "Written  Exercises. 
Combine  when,  possible : 

1.  3  abc2  —  abc  +  abc2  =  4  abc2  —  abc. 
The  first  and  third  terms  are  combined. 

2.  3  abc2  —  ab2c  +  a2bc. 

3.  3  abc  —  2  abc  +  abc. 

4.  2  abx  —  ab  -f  3  bx. 

5.  a3_a2  +  2a-4  +  3a2-a. 

6.  ±a2-2ab  +  b2  +  a2-b2. 

7.  8  arfy  —  xy  -f-  a?  —  xy  —  3  #. 

8.  4  ra2  +  4  ran  +  n2  —  mw2. 

573.  Written  Exercises. 
1.    (a2  +  &)(a  +  6)  =  ? 

Multiplying  by  a,  we  have  a3  +  ab.     Multi-       a2  +    5 
plying  a2  by  6,  gives  a26.     As  there  is  no  like       a  +    b 


term,   this  product  is  placed  after  ab.     The       az  +  ^&  -f  a26  +  b2 
product  of  &  by  6  is  placed  last. 

Rearrange  the  terms  in  the  order  of  the  size  of  the  exponents  of  a. 
Ans.   as  +  a?b  +  ab  -f  62. 

2.  (2a&- &)(&-l)  =  ?  5.    (a2-62)(a  +  &)  =  ? 

3.  (2a&-&)(2a-l)  =  ?  6.    (a  +  b)(c+d)  =  ? 

4.  (2a&-&)(a&-&)  =  ?  7.    (3a  +  b)  (a-  a&)  =  ? 

8.  (2j92-a)(2pg  +  a;)  =  ? 

9.  (x*  +  ±y2)(±x-y)  =  ? 
10.    (^  +  «-2)'(a;-2)=? 


456  Chapter  Seven. 

DIVISION. 

574.  Preliminary  Exercises. 

1.  6  a  -r-  3  =  2  a.  4.   5  a2 a;  -f-  5  a2  =  jc. 

2.  a2  -r-  a  =  a.  5.   10 a2 a;  -7-  5 a;  =  2  a2. 

3.  6a2-j-3a=2a.  6.   10  a2x  ■+■  2 ax  =  5  a. 
7.  (12a3a;?/2-27aVi/)^3a2a;2/  =  4a?/-9a;. 

In  the  above  examples  what  is  done  with  the  coefficients  ? 

What  is  done  with  the  exponents  of  the  same  letter  ? 

When  the  divisor  is  a  monomial,  divide  the  numerical  coeffi- 
cient of  each  term  of  the  dividend  by  the  numerical  coefficient 
of  the  divisor.  Then  write  the  letters  of  the  dividend,  giving 
each  an  exponent  equal  to  the  exponent  in  the  dividend  dimin- 
ished by  that  in  the  divisor. 

575.  Like  signs  give +.     Unlike  signs  give  — . 

576.  Written  Exercises. 
Divide : 

1.  5  a3  by  a2.  4.    —  xPy3  by  —  xy2. 

2.  —  xPy3  by  xy2.  5.    x2  +  2  xy  by  x. 

3.  xPy3  by  —  xy2.  6.   Xs  —  5  x2  +  3  x  by  x. 

7.  15  rf4  -  10  a8  +  20  a?"  by  -5a2. 

8.  4  a3?/3  —  3  x2y  +  #?/2  by  xy. 

9.  -12afy2  +  33.'c2?/3-24<c3/4by  -3^. 

10.  —  7  a5  by  —  7  a4. 

11.  7  a5  by  —  7  a5. 

12.  a2  +  a  by  J  a.  -4ns.  2  a  +  2, 

13.  oj8  +  3  05*  by  — -J  a?. 

14.  6  a3  -  3  a2  by  -  9  a2. 

15.  —x  +  xy  —  xzby  —x. 

16.  £afy-2ayby  -*afy. 


Algebraic  Equations.  457 


577.  Written  Exercises. 
1.   Divide  x2  +  7  x  +  12  by  x  +  3. 
We  write  a  division  like  the  above  as  follows  : 


x2  +  7  x  +  12 


3  +  3 


The  first  term  in  the  divisor  is  x. 

The  first  term  in  the  dividend  is  32. 

Dividing  x2  by  x  we  get  x  for  the  first  term  of  the  quotient,  which 

3  +  3 
x 


we  put  down  thus,  32  +  7  3  +  12 


then  we  multiply  the  divisor  by  the  first  term  of  the  quotient  and  write 
the  result  under  the  dividend. 

x2  +  7  x  +  12  I  3  +  3 
(re  +  3)  x  times  x2  +  3  x  \x 

Next,  we  subtract  x2  +  3  x  from  the  dividend, 

x2  +  7  x  +  12  |  a;  +  3 
subtract  a2  +  3  a;  |  x 

remainder  4  x  +  12 

Next,  we  divide  the  first  term  of  the  remainder  by  the  first  term  of 
the  divisor  and  get  +  4,  which  we  write  in  the  quotient  thus, 


x2  +  7  x  +  12 

x2  +  Zx 

3  +  3 
3  +  4 

43  +  12 
Next,  we  multiply  the  divisor  by  4  and  write  the  product  thus, 

3  +  3 


X2  +  7  x  +  12 
32  +  3  3 

43+12 
(3 +  3)  4  times  43  +  12 


3  +  4 


Subtracting,  there  is  no  remainder.     The  whole  quotient  is  x  +  4. 

32  +  7  3  +  12  I  3  +  3 

32  +  33  I  3  +  4     An8. 

43  +  12 
4  3  +  12 


45  8  Chapter  Seven. 

2.   Divide  a^  +  lSx  +  56  by  m  4-4. 
x2  +  18  x  +  56  I  x  +  4 


x2  +    4x 

14  x  +  56 

14  a;  +  56 

0 

|  se  +  14    u4/w. 

3.   Divide  a2-2ab~ 

2462  by  a-f  46. 

a2  -  2  ab  -  24  62  1 
a2  +  4  a6 

-  6  a6  -  24  62 

-  6  a6  -  24  62 

a  +  46 

a  —  6  6    Ans. 

0 
Prove  that  the  answers  in  the  above  examples  are  correct. 

578.   Written  Exercises. 
Divide : 

1.  x*  +  5x  +  6  by  x  4-3. 

2.  aj84-5o2  +  6a;  by  a?  4- 3. 

3.  ar*  4- 5  a?  4- 6  a?  by  x2  +  3x.  Ans.  x  +  2. 

4.  a^+7a;2/4-102/2by  x  +  2y. 

5.  a2  — 7a#  +  10y*by  a?-2y. 

6.  3aj2  +  14ajy  +  82/8byoj  +  4y. 

7.  3a^  +  10a;2/-8^2by  »  +  4y. 

8.  3a^-10^-8?/2by  aj-4?/. 

9.  3052  —  14  an/ 4- 8  ?/2  by  a;  —  4y. 

10.  3^  +  10^  —  8?/2  by  3a  —  2y. 

11.  8^  +  22a;?/  +  152/2by  2»  +  3y. 

12.  8a?  — 2ajy-15y*by  4x  +  5y. 

13.  8a;2-22a;?/  +  15?/2by4a;-5y. 

14.  n¥  +  3  ana?  +  2  a2  by  wz  4.  a.         ^4ws.  no;  4-  2  a. 

15.  n¥  4-  anx  —  2a*  by  nx—  a. 

16.  6a2b2  —  13dbw  +  6w2by  3ab-2iv. 

17.  4^4-2a*/z-1322/2*2by  2x-llyz. 


Algebraic  Equations. 


459 


18.   ±ax*  +  2axyz  —  132ai/Vby  2x  —  11  yz. 
Id.   8  ax2  —  26  axyz  -f  15  ay2z2  by  4  cue  —  3  ayz. 


579. 

Written  Exercises. 

l. 

Divide  Xs  ■ 

-1 

by  x 

— 

1. 

Xs 

-l  | 

X 

-1 

0 

-32 

X2 

+  X+1 

X* 

-  1 

X2 

—  X 

X  — 

1 

X  — 

1 

Ans. 


2.   Divide  Xs  -  13  x  -  10  -  2  or2  by  a;  -  5. 

The  terms  of  the  dividend  should  be  arranged  according  to  the  size 
of  the  exponents  of  £,  thus,  xz  —  2  x2  —  13  x  —  10. 


a8  -  2  z2  -  13  re  - 

9 

x*-bx2 

3  a2  -  13  x  - 
3z2-15a; 

9 

x2  +  3  a;  +  2  + 


x-5 


S«-    9 

2s-10 

+    1 


Divide : 

3.  a3  —  63  by  a  —  6. 

4.  cs-lby  c-1. 
7.    c3  +  2  by  c  -f- 1. 


5.  c8  +  1  by  c  4- 1. 

6.  9a4-1662by  3a2+46. 

^4ns.  c2  +  c  +  1  +  -^— • 
c  +  1 


8.  ar5  +  6afy  +  :12a;2/2  +  8?/3by  x  +  2y. 

9.  a3  +  3  a  —  3  a2  —  1  by  a  —  1.     Rearrange  dividend. 

10.  or3  +  27  ^  by  x  +  3  y. 

11.  a2  +  2a&  +  &2-l  by  a  +  &  +  l.  ^Ins.  a  +  &-l. 

12.  12a3-20a2  + 33a-5by  6o-l. 

13.  8  a&3- 125  aafy3  by  2ab-5axy. 

14.  l  +  5a3-6a4by  l-a  +  3a2.     * 

15.  6a5  +  a;-12^  +  9ic3-3-lla;4by2»3-3^-l. 


460  Chapter  Seven. 

FACTORING. 

580.  The  expression  x  +  xy  may  be  divided  by  x ;  the 
quotient  is  1  +  y.  The  factors  of  x  +  xy  are  x  and  1  +  y ; 
that  is,  (#  +  xy)  =  se(l  +  ?/). 

In  a  similar  manner  we  find  that  the  factors  of  2  a2b  +  4  a&2 
are  2  a&  and  a  -f  2  6 ;  that  is,  2  a26  +  4  a&2  =  2  ab  (a  +  2  6). 

581.  Factor: 

1.  m  +  mn.  4.   a?  +  a\  7.   c3  +  c2  +  c. 

2.  m2/i  +  2m.  5.   6  63-962.  8.    7  a26  +  21  a2c. 

3.  6afy-3a?/2-  6.    a36  +  6a&3.         9.   4x4+6a3. 

10.  4x4  +  6arJ  +  8x?.  18.  20 ax2  -  15 bx3  +  16 ex4. 

11.  4x4  +  6arJ  +  8x2  +  10a>.  19.  20ax2  +  15a2ar}-20a3a;4. 

12.  4z4+6ar5+8z2+10x+12.  20.  a¥  +  aV. 

13.  ^a^-gacr'+eac3.  21.  6  a?  +  1  a*  +  2  a«. 

14.  3a62  +  2a26  +  2a3.  22.  a2xA  +  a3^3  +  a4s4. 

15.  3a26  +  6a&2-15a&8.  23.  12  m3n  -f  5 m2n2  + 15  ny. 

16.  #yz  +  xyz2.  24.  a7  —  a562  -f  a4^. 

17.  9  m%  -  27  msn2y.  25.  70  a7  +  60  x6  -  50  a5. 

582.  The  square  of  x  -f-  y  is  x2  -f-  2  xy  -f  y\ 

Note  that  x2  and  y2  are  the  squares  of  x  and  y,  respectively, 
and  that  2  scy  is  twice  the  product  of  x  and  y. 

The  square  of  the  sum  of  two  quantities  is  equal  to  the  square 
of  the  first,  plus  twice  the  product  of  the  first  and  the  second, 
plus  the  square  of  the  second. 

Any  expression  in  the  form  of  x2  -f  2  xy  -f-  y2  is  composed 
of  two  equal  factors. 

a2  +  2a&  +  &2  =  (a  +  6)(a  +  b)  or  (a  +  &)*. 


Algebraic  Equations.  461 


583 

1    Factor: 

1. 

c2  +  2cd  +  d?. 

14. 

c2d246cdm49m!. 

2. 

m2  4  2  m  +  1. 

15. 

4a2  +  12a  +  9. 

3. 

4  +  4w  +  m;2. 

16. 

9a2  4- 12  ab  4  4  6*. 

4. 

a¥  4  2  axy  4  y2. 

17. 

4  a2  4-  4  ac  4-  c2. 

5. 

r2  +  2  rs£  4  s2*2. 

18 

a*4-2a2&4&2. 

6. 

e2  +  6e+9. 

19. 

a2  4  2  abx  4  &W 

7. 

x2  +  8  a;  4  16. 

20. 

a262  4  2  afcca"  4  c¥. 

8. 

4  624  46d4d2. 

21. 

m2n24l0mn  +  25. 

9. 

a2  4  4  ay  4  4  y2. 

22. 

962  +  30  6c4  25c2. 

10. 

a2  4  4  ayz  4  4  ?/222. 

23. 

16  4l6a*/4-4afy2. 

11. 

4a2z2  +  4a6a;4&2. 

24. 

x4  4  2  afy2z  4  2/V. 

12. 

M*4-6ttv  +  9v*. 

25. 

a2624-12a6cH-36c2. 

13. 

a44-2a2624&4. 

26. 

4  ra2  +  8  mw  -|-  4  w2. 

584.  The  square  of  a  —  b  is  a2  —  2  a&  4  62. 

Compare  this  form  with  that  of  Article  582,  and  note  the 
difference  in  signs. 

Give  a  general  statement  for  the  square  of  the  difference 
of  two  quantities. 

585.  Factor: 

1.  a2- 2 ay  +  if.  14.   x2-2x  +  l. 

2.  l-2x  +  x*.  15.    4z4-4arJ4l. 

3.  m2-2mnr  +  w¥.  16.    4  x2  —  12  xy  4  9  y2. 

4.  a2-4a4  4.  17.    9  y2  -  12  xy  4  4ar>. 

5.  9-6&4&2.  18.    a4  -  2  a2a  4  ar>. 

6.  rV  -  2  rs£  4  Z2.  19.    9  -  12  b  4  4  62. 

7.  c*-4  cd4  4d2.  20.    4^44  a;  41. 

8.  16  b2-  8  a;  +  1.  21.    x2  +  4  y2  -  A  xy. 

9.  4  b2-  4  6c  4  c2.  22.    16  x2  -  40  a;z  4  25  z*. 

10.  a2?/2  -  4  ayz  4  4  z2.  23.    b4  -  2  b2cd2  +  c2d\ 

11.  4  a2?/2 -4  ayz +  z2.  24.    25  z4  -  30  x2  4  9. 

12.  9a2-6ah  +  h2.  25.    25  ar*  -  30  ar*  4  9  a. 

13.  62c2-66cd49d2.  26.    503^-60^4183?. 


462  Chapter  Seven. 

586.   The  product  of  a  +  b  and  a  —  b  is  a2  —  62. 

Give  a  general  statement  for  the  product  of  the  sum  and 
the  difference  of  two  quantities. 

An  expression  consists  of  the  difference  of  the  squares  of 
two  quantities.     What  are  the  factors  of  the  expression  ? 

In  factoring  an  expression  first  examine  it  to  see  if  it  contains  a 
monomial  factor.  o2  —  be2  =  b  (b2  —  c2).  The  factors  of  b2  -  c2  are 
6  +  cand6-c.     63  -  be2  =  b  (b  +  c)(b  -  c). 


12.  m2-9n2. 

13.  b5-9b. 

14.  9  m2  —  n2p2. 

15.  9  62-4. 

16.  4  c2  -9^. 

17.  6c2-4  6d2. 

18.  aA-b2. 

19.  x*-y2. 

20.  xif  —  a2b2x. 

21.  afy2-a262. 

22.  -m2  +  aV. 

588.    The  product  of  x  +  2  and  a?  +  3  is  ar2  4-  5  a;  +  6. 

Note  that  the  coefficient  of  the  second  term  of  the  product 
is  the  sum  of  the  second  terms  of  the  factors ;  5  =  2  4-  3. 

Note  that  the  last  term  of  the  product  is  the  product  of 
the  second  terms  of  the  factors ;  6  =  2  times  3. 

Factor  x2  +  7  x  + 12. 

We  must  find  two  numbers  whose  sum  is  7  and  whose 
product  is  12.     These  numbers  are  3  and  4. 
a?  +  7  x  +  12  =  (x  +  3)  (x  +  4). 

In  a  similar  manner,  we  find 

x2  +  6  x  +  5  =  (x  +  1)  (a?  +  5). 


587.   Factor: 

1. 

m2  —  n2. 

2. 

1-m2. 

3. 

a^-4. 

4. 

c2  -  ay. 

5. 

9-v2. 

6. 

h2  - 16. 

7. 

4  ar*  —  y2. 

8. 

oV  —  #2. 

9. 

a2  —  4  ?/2. 

10. 

4  a262  —  w2. 

11. 

aw2  —  4  a. 

Algebraic  Equations.  463 

589.  Factor: 

1.  ^  +  43  +  3.  14.  7*  +  10r  +  9. 

2.  m2  +  6m  +  8.  15.  s^lls  +  lS. 

3.  62  +  7&  +  10.     \  16.  24  +  10a  +  a2. 

4.  c2  +  7c  +  6.  17.  ^  +  11^  +  24. 

5.  a2  +  8a  +  7.  18.  20  +  12^  +  ^. 

6.  a2  +  9a  +  14.  19.  x2  +  12x  +  32. 

7.  y*  +  8y  +  12.  20.  27  +  12x  +  x*. 

8.  d2  +  10d  +  16.  21.  a2  +  14a  +  24. 

9.  7t2  +  87i  +  15.  22.  c*  +  20c  +  19. 

10.  a2 +  9  a; +  18.  23.  2/2  +  12?/  +  35. 

11.  a;2  +  9a;  +  20.  24.  ra2  +  13m  +  30. 

12.  z2  +  10z  +  21.  25.  ar'  +  llaj  +  SO. 

13.  w2  + 12w  +  20.  26.  /2  +  10/+9. 

590.  The  product  of  x  —  2  and  a;  —  3  is  ar*  —  5  a;  +  6. 
Compare  with  Article   588  and  note   the   difference  in 

signs.  a2  - 11  a  + 10  =  (a  - 10)  (a  - 1). 

591.  Factor: 

1.  c2-7c+12.  9.  w2-14w  +  45. 

2.  <z2-5d  +  4.  10.  n2-18n  +  45. 

3.  a2-13a;  +  22.  11.  ^-16^  +  28. 

4.  2/2-142/  +  33.  12.  36-15z  +  z2. 

5.  22-12z  +  ll.  13.  &2-17&  +  30. 

6.  a2-13a  +  40.  14.  30-316  +  62. 

7.  m2-18m  +  32.  15.  s?-16s  +  55. 

8.  w2-14n  +  13.  16.  y2-16y  +  63. 


464  Chapter  Seven. 

17.  48-14c  +  c*.  22.  80  +  18?/  +  2/2- 

18.  z2-27z+50.  23.  d2  +  17d  +  72. 

19.  tv2  +  20w  +  51.  24.  a^-20a;  +  99. 

20.  50-fl5*  +  *2.  2*.  99-100a  +  a2. 

21.  h2  +  17h  +  4:2. 

592.    1.  The  product  of  x  +  5  and  x  -  3  is  a2  +  2  a;  -  15. 
2.   The  product  of  ic  —  5  and  x  +  3  is  or2  —  2  re  —  15. 

What  gives  a  minus  sign  in  a  product  ? 

Why  is  the  sign  of  the  last  term  of  the  product  minus  in 
each  of  the  above  statements  ? 

Note  that  in  each  of  the  products  the  coefficient  of  the 
second  term  is  the  difference  between  5  and  3. 

In  the  factors  in  the  first  statement,  has  the  larger  num- 
ber a  plus  or  a  minus  sign  ?  What  is  the  sign  of  the  second 
term  of  the  product  ? 

In  the  factors  in  the  second  statement,  has  the  larger 
number  a  plus  or  a  minus  sign  ?  What  is  the  sign  of  the 
second  term  of  the  product  ? 

Factor  a^-4«- 45. 

We  must  find  two  numbers  whose  product  is  45  and  whose 
difference  is  4.     We  see  that  9  and  5  are  such  numbers. 

The  sign  of  the  second  term  in  the  given  expression  is 
minus,  so  we  must  give  the  minus  sign  to  the  larger  of  the 
two  numbers  which  we  have  found. 

rf  _  4  x  -  45  =  (x  -  9)  (x  +  5). 

In  a  similar  manner,  we  would  get 

x2  +  4  x  -  45  =  (x  +  9)(x  —  5). 
x2  +  x-56  =  (x  +  S)(x-7). 
x2-x-56  =  (x-$)(x  +  7). 


Algebraic  Equations.  465 

593.  Factor: 

1.  a^-x-12.  14.  ar* -5 a; -36. 

2.  a*  +  x-12.  15.  a2-5a-14. 

3.  a2  +  3a_io.  16.  b2-7b-S0. 

4.  a2_3a_io.  17.  z2-2z-35. 

5.  2/2_42/_i2.  18.  ra2  +  4m-60. 

6.  y*  +  4y-12.  19.  &2  +  3&-54. 

7.  c2  +  5c_6.  20.  x2  +  2x-4,S. 

8.  d2-7d-18.  21.  *2-10*-ll. 

9.  d2  +  7d-18.  22.  ?/2  +  7  ?/ -  60. 

10.  k2-7k-$.  23.  /i2-9/i-36. 

11.  02_20_24.  24.  a2-2a-120. 

12.  a2 -5  a; -24.  25.  d2  +  5  d  -  150. 

13.  ^  +  5^-36.  26.  d2-5d-150. 

594.  If  we  divide  2?  —  y3  by  x  —  y,  the  quotient  is  ar*  +  xy 
+  2/2-  a3_  53  =  (a  _  6)(a2  +  ab  +  j^ 

The  difference  of  the  cubes  of  two  quantities  may  be 
divided  by  the  difference  of  the  quantities.  The  quotient 
consists  of  three  terms,  namely  :  the  square  of  the  first,  plus 
the  product  of  the  first  and  second,  plus  the  square  of  the 
second.  (as  +  6sj  +  (a  +  fy  =  a2_a0  +  62# 

Give  a  general  statement  for  the  quotient  obtained  by 
dividing  the  sum  of  the  cubes  of  two  quantities  by  the  sum 
of  the  quantities. 

Factor  a3  4-  8.  8  =  23 

hence  a3  +  8  =  a3  +  23, 

a3  +  23  =  (a  +  2)  (a2  -  2  a  +  22), 

since  22  =  4, 

a3  +  8  =  (a  +  2)  (a2  -  2  a  +  4). 


56 

Chapter  Seven. 

595.   Factor: 

1.   ra3  —  x5. 

10.   m3-f-8n3. 

18. 

a*-tf. 

2.    m3  +  ar*. 

11.   a363  +  8. 

19. 

7^  +  27. 

3.    a3  +  l. 

12.   8yV  +  l. 

20. 

a3 -64. 

4.    a3-l. 

13.    (3x)s-f. 

21. 

a3  +  64. 

5.    1-a3. 

14.    ar>+(3)3. 

22. 

a?  -  64  y5. 

6.    &-<*&. 

15.   l  +  27d3. 

23. 

a363  +  <W. 

7.    fcV+r3. 

16.   27  a3 -1. 

24. 

64 -z3. 

8.    (2a)3-63. 

17.    (a2)8-63. 

25. 

a3  - 125. 

9.    S^-c8. 

596.  The  following  supplementary  exercises  are  applica- 
tions of  the  preceding  cases.  Sometimes  it  is  possible  to 
apply  more  than  one  case  to  an  exercise. 

1.  Factor  2  (fix  —  2  a¥-  24  a5. 
First  divide  by  2  x. 

2a4x  -2a2x*-  24  a5  =  2a(a4  -  aV  -  12a4), 
but  a4-aV-  12a4  =  (a2  +  3ar2)(a2  -  4^); 

hence  2  a4*  -  2  aV  -  24  a^>  =  2  x(a2  +  3  ar2)  (a2  -  4  ar2), 
but  a2  -  4  ar>  =  (a  +  2  a?)  (a  -  2  ar) ; 

hence  2a4a;-2cW-  24rf  =  2aj(a2  +  3«8) (a  +  2 a?) (a -2 a). 

2.  Factor  ai6  — #6. 

Since  xe=(xi)2,  and  ^=(jf)29 

x*-tf=(x*  +  tf)(7*-tf). 
Factoring  the  second  member  of  the  above  equation, 
**  —  f  =  0*  +  V)  («*  -  ^  +  2Z2)  (*  —  2/)  0*2  +  xy  +  t/2)  . 
When  in  doubt  prove  your  work  by  either  multiplication 
or  division. 


Algebraic  Equations.  467 

597.   Factor: 

1.  7  s?y  + 21  xyz2.  27.  Sb5-5b\ 

2.  ab2-2abc  +  ac2.  28.  (z  +  ?/)2-l. 

3.  a2t>  +  3a&c  +  2  6c2.  29.  a2-2 ab  +  62-l. 

4.  a3  +  2a2  +  a.  30.  z2-*5. 

5.  2^-216.  31.  125  x  +  x\ 

6.  a?b2c?  +  6abc  +  9.  32.  <e4-5a:2  +  4. 

7.  3n4  +  81tt.  33.  a4-2a2-8. 

8.  n4-81w.  34.  100  + 25  ?/2. 

9.  a4 -a.  35.  100-  25  y2. 

10.  2a2-2a&-246*.  36.  ab2  - 144  ac2. 

11.  a4  +  2cc22/2  +  y4.  37.  12  a*/2  - 144  xz\ 

12.  a^  +  5x.  38.  3z2-9a-84. 

13.  tf  —  25tf.  39.  a^c2  —  a  +  2  a&x  +  ab\ 

14.  m2  +  25  mn  + 100  n2.  40.  a4 +2  ar'  +  l -a2. 

15.  a2  -  20  a&c  + 100  W.  41.  z4  +  2  ar>  +  l  -s2. 

16.  ^-16^.  42.  a4  +  a2  +  l. 

17.  a4  —  16  a26  +  55  b2.  (Change  42  to  same  form  as 

18.  o^  +  aY-  41° 

19.  z4  +  *y.  43'  2/2- 13  2/ +  36. 

20.  *4-*y.  44'  ^"13  62  +  36. 

21.  fl/-^f-'6«  45'  *+'*+* 

22.  ^-5^2  +  6^.  46'  ™4-16. 

23.  a362  +  5a263  +  10a65.  47.  16-wiW. 

24.  (3m)2  +  2(3m)  +  l.  48.  16m  +  2m4. 

25.  (x  +  y)2  +  2(x  +  y)  +  l.  49.  16  a -a4. 

26.  (x  +  y)2-2(x  +  y)z  +  z2.     50.  a6 -64. 


468  Chapter  Seven. 

FRACTIONS. 
598.   Preliminary  Exercises. 

1.  Keduce  9!  to  its  lowest  terms.  Ans.  ^-> 

xz  z 

Divide  the  numerator  and  the  denominator  by  x. 

2.  Change  ^  to  an  equivalent  fraction  whose  denominator 

is  be.  Ans.  ^ 

be 
Multiply  the  numerator  and  the  denominator  by  c. 

The  value  of  a  fraction  is  not  changed  when  both  numerator 
and  denominator  are  either  multiplied  or  divided  by  the  same 
quantity. 


599.   Oral  Exercises 

Eeduce : 

1.    $?*.              2. 

3x 

2a& 

o 

7  b  h?                    7  a3 
21  &W                     a3 

3a2' 

ax-\-  ay 

7. 

2ab2 

9     nxy  +  a; 

Sab 

a-\-  ab 

ax-f-x2 

6     12a*b\ 

8. 

a2  +  2ab 

10    2px  +  3xy 

6  ab 

3ab 

px  +  xy 

600.   Sight  Exercises. 

Give  answers  at  sight : 

,     b       ? 

a         16  a3 

1.   -  =  — . 

6. 

4a  =  — - — 

3     3x 

? 

2     2V-    ?  . 
'    3x     3a2 

7. 

2/  +  *_    ?   . 
2  a       2a& 

2a     2ac 

8. 

2^-2f- 

'     3  ~    ?  " 

? 

4     5a*/_      ?     . 

9. 

3  7^  =_J 

7  ra      14  ran 

3  rat 

5.   4a  =  -. 

a 

10. 

.  .  »      2a2&  +  2a# 
a  +  0  = - 

Algebraic  Equations.  469 

601.  Written  Exercises. 

,     .p   ,        a?  +  lire +  30 
1.   Keduce  — -*- -1 

x  +  5 

Divide  the  numerator  by  the  denominator.  Ans.  x  +  6. 

a.  Keduoe  *+i2»+35. 

x  +  5 


3.    Keduce 


^  +  10a;  +  20.  Jjm.  a +  6 


a;-f4  #  +  4 

1  2c  +  3 


6.    Change  x+2  to  a  fraction  whose  denominator  is  x+3. 
•  _j.  2  =  ?-i —    Multiply  the  numerator  and  the  denominator  by  x + 3. 


7.    a  -3  =  —^—.  8.    a^-ic  +  l  = 


x  +  5  a+1 

a  f  racti 

3^  +  23; 


x  —  3 

9.   Change  a;  H to  a  fraction. 

x-\-  2 


x  — 


x  +  2 


x*  +  2x     x-3 


x  +  2       x  +  2     x  +  2 
What  may  be  done  with  the  numerators  when  the  denominator  is 


common 


10.   Eednce  **-}.  13.   -^+3aJ 


2x-l  x+2     x+2 

11.    x  —  1 -  = -•  14.    -  =  • 


z  +  2     x  +  2  x     2x*  +  x 

3  5?  2         9 

12.    7^-  +  —^  =         '    - —        15. 


2x     x  +  2     2x?  +  4:X  y     y*  —  y 

16.  §+•=«  +  *-« 


a;     2a;  +  l      2^  +  a;     2^  +  a; 


470  Chapter  Seven. 

"•   [x-l)[x-2j     ■'    e-'  (x-l)(x-2)      C 
|  multiplied  by  f  =  ? 

18     ^±1^^L?  =  9.  u    (a;  +  1)^-2)==? 
*   a_i  *  a;_2      *'   .   '  («-l)(a  +  2)      ' 

19.   By  what  quantity  must  a;  —  5  be  multiplied  to  give  a 
product  of  a2 +  z -30? 

By  what  number  must  7  be  multiplied  to  give  a  product  of  63  ? 
1  ? 


20. 
21. 
22. 
23. 


x-5     ^  +  05-30 

a  ? 

a;_6     a^_a;-30* 

a        aac-f»q 

a-l~~? 

x-2\fx  +  5\  =  l> 
x-5j\x  +  2j      ' 


24. 

05  +  1      a?  +  2x  +  l 
x-1              ? 

25. 

x-5         ? 

s  +  1     a^-l 

26. 

a;  — 5  ,  x  —  1      9 

a;  +  3  '  a;-h4- 

27. 

Add*-*and*-?. 
a;  -}- 1         a— 1 

Ans.  2^-7»  +  8 
as»-l 

28. 

a;  — 2     x  —  5  _  9 
a;  — 1     a;-f  1 

i 

29. 

z  +  5_         ? 

a;_3     3^-^-6 

30. 

27a8a?-2_? 

3CUB-1 


Algebraic  Equations.  471 

PURE  QUADRATICS. 

602.  Given  ^±-^  =  3a?2~66,  to  find  the  value  of  x. 

5  9 

Clearing  of  fractions,  9  z2  +  54  =  16  z2  -  330. 
Transposing  and  combining,  —  6  z2  =  —  384. 

Dividing  by  -  6,  x2  =  64. 

Extracting  square  root,  z  =  ±  8. 

Since  (—8)  x  (—  8)  =  64,  the  square  root  of  64  may  be 
either  +  8  or  —  8.  It  is  written  ±  8,  and  is  read  "positive 
or  negative  8."  (It  is  sometimes  less  correctly  called  plus  or 
minus  8.) 

603.  Written  Exercises. 
Find  value  of  x,  y,  z,  etc. : 

1.  ^-13  =  36. 

2.  3/ +25  =  100. 

3.  5z2-13  =  3z2  +  37. 

4.  5(ar}-fl7)-3arJ+63  =  198. 

5.  5(^  +  17)  -3(^-21)  =198. 

6.  2/2  +  22/  +  l-2/2  =  49. 

7.  (a;-f-l)2-arJ=49. 

8  i/2  +  5     2y2-18_o 

8.  — 4— 2' 

9  z  +  7  =  z-5_ 
z_3     z_9* 

10      20a;  =  30a; 
x  —  1     a;  +  1 


472  Chapter  Seven. 

11.  (a; -3)  (a? +  3)  =  40. 

12.  (x  +  5)(x  +  5)=10x  +  26. 

13.  (x  +  4:)2=8x  +  80. 

14.  z2  +  64  =  5z2. 

15.  3^  +  18  =  21^  +  36. 

16.  (x-3)2-(x-5y  =  12. 

17.  (aj  +  7)(aj-9)  =  (aj-3)(a?-6). 


18. 

4      a; 

X 

5 

+5- 

a; 

19. 

«  +  7 
a;  —  5 

a; 

a; 

-3 
-9 

20. 

2/-9_ 
y  —  5 

2/ 

-3 

+  7 

604. 

"Written  Problems. 

1.  Find  the  dimensions  of  a  field,  the  length  of  which  is 
twice  its  breadth,  its  area  being  1800  square  rods. 

2.  The  surface  of  the  six  equal  faces  of  a  cube  contains 
96  square  inches.     Find  the  length  of  one  edge. 

3.  One  number  is  four-fifths  of  another,  and  their  product 
is  80.     What  are  the  numbers  ? 

4.  One-third  of  a  number  multiplied  by  two-fifths  of  the 
same  number  gives  a  product  of  270.     Find  the  number. 

5.  Thirty  per  cent  of  a  number  multiplied  by  forty  per 
cent  of  the  same  number  gives  a  product  of  300.  What  is 
the  number? 

6.  Thirty  per  cent  of  twenty  per  cent  of  a  number  is 
300.     What  is  the  number  ? 


Algebraic  Equations.  473 

7.  The  base  of  a  right-angled  triangle  is  f  as  long  as 
the  perpendicular,  and  the  area  of  the  triangle  is  96  square 
rods.  Find  the  length  of  the  base.  What  is  the  length  of 
the  hypotenuse  ? 

8.  The  base  of  a  right-angled  triangle  measures  x  yards, 

3  x 

the  perpendicular  measures  —  yards.     What  is  the  length 

of  the  hypotenuse  ?     If  the  hypotenuse  measures  15  yards, 
find  the  length  of  the  base. 

9.  The  base  of  a  right-angled  triangle  measures  x  feet, 
the  hypotenuse  measures  (x-\-  9)  feet,  the  perpendicular 
measures  15  feet.     What  is  the  length  of  the  base  ? 

10.  The  difference  between  the  squares  of  two  consecutive 
numbers  is  49.     What  are  the  numbers  ? 

11.  The  difference  between  two  numbers  is  6.     The  sum 
of  their  squares  is  146.     What  are  the  numbers  ? 

Let  x  —  3  =  smaller  number, 

and  x  +  3  =  greater  number. 


AFFECTED   QUADRATICS. 
605.   Preliminary  Exercises. 

(x  + 1 )  (x  +  1 )  =  z?  +  2  x  + 1 . 
Compare  with  Article  582. 

0-l)(a;-l)=a;2-2a;  +  l. 
Compare  with  Article  584. 

(a  +  6)2  =a2  +  2a&  +  62. 
(m  —  n)2  =m2  —  2  mn  -f-  n2. 
(10  +  5)2=  102  +  2  x  10  x  5  +  52. 
(10-3)2  =  102-2xl0x3  +  32. 


474  Chapter  Seven. 

606.  Oral  Exercises. 
Square : 

1.  3J  +  3.  4.   3J  +  10.  7.    30  —  1.  10.    x-y. 

2.  *— 7,  5.    a-6.  8.   40-1.  11.    80  +  5 

3.  x  —  9.  6.   #  +  y.  9.   m  +  n.  12.   60  —  5. 

607.  Oral  Exercises. 
Extract  the  square  root  of 

1.  a^  +  6a;  +  9.  6.  x2  +  2xy  +  y2. 

2.  a2-14a;  +  49.  7.  a? - 2 an/  +  ?/2. 

3.  ar>-18a;  +  81.  8.  a2-2ab  +  b2. 

4.  a^  +  20a;  +  100.  9.  a2  -  24  x  + 144. 

5.  a2  +  2a6  +  62.  10.  z2  +  22 a; +  121. 

The  square  of  (x  +  3)  consists  of  how  many  terms  ?  Of 
how  many  terms  does  (x  +  4)2  consist  ?    (a;  +  5)2  ? 

608.  Supply  term  necessary  to  make  a  complete  square : 

1.  x*  +  6x+?  6.  x*  +  2x  +  ? 

2.  a^-12a;  +  ?  7.  3^-43;+? 

3.  3^-83;  +  ?  8.  «2-10a;  +  ? 

4.  3^-163?+?  9.  3^+143;  +  ? 

5.  3^  +  183;  +  ?  10.  x2-22a;  +  ? 

609.  "Written  Exercises. 

Given  x2  +  6x  =  27. 

What  number  must  be  added  to  the  first  member  of  the 
equation  to  make  it  a  "  complete  "  square  ? 

If  a  number  is  added  to  one  member  of  an  equation,  what 
must  be  done  to  the  other  member  to  preserve  the  equality  ? 


Algebraic  Equations.  475 

610.  Extract  the  square  root  of  both  members  of  the  fol- 
lowing equations,  adding  to  both,  where  necessary,  such  a 
number  as  will  make  the  first  member  a  complete  square. 

1.   x*  +  6x  +  9  =  40  +  9.         2.    a?—12x  +  36  =  28  +  36. 
Remember  that  (+  7)  x  (+  7)  =  49,  and  that  (-  7)  x  (-  7)  =  49. 
.-.   V49  =  +  7  or  -  7,  written  ±  7. 

3.  ^-8^  +  16  =  20  +  16.  7.  »*- 14 a; =15. 

4.  arJ-16a  +  64  =  -39  +  64.  8.  ar»-22a;  =  23. 

5.  <c2  +  18a;  +  ?  =  19  +  ?  9.  a" +  14 a  =  51. 

6.  jb2  +  2oj+?  =  24  +  ?  10.  x2-22x  =  AS. 

611.  Given  a2 -10*  =  24. 

Completing  the  square,  we  have  x2  —  10  x  +  25  =  24  +  25  =  49. 
Extracting  the  square  root  of  both  sides,  we  have 

x  -  5  =  ±  7, 

x  =  7  +  5  =  12,  orx  =  -7+5  =  -2. 
Ans.   12  or  -  2. 

612.  Written  Exercises. 
Find  values  of  x : 

1.  x*-6x  =  7.  9.  ar*  -  24a  =  0. 

2.  ^-12^  =  108.  10.  ar2-8a;  =  384. 

3.  or*  +  2a;  =  48.  11.  a?2  — 4a;  =  —  3. 

4.  a2 +  18  a;  =  115.  12.  ar>  +  30a;  =  175. 

5.  a^-14a;  =  -13.  13.  ar>  +  28a;  =  29. 

6.  ^-10a;  =  0.  14.  x2  +  22 x  =  104. 

7.  x>  +  20x  =  125.  15.  x2-16x  =  -6±. 

8.  arJ  +  26a;  =  56.  16.  ar5  +  36a;  =  76. 

To  make  the  first  member  a  complete  square,  you  added 
the  square  of  what  part  of  the  coefficient  of  x  ? 


476  Chapter  Seven. 

613.  Written  Exercises. 
Find  values  of  x : 

1.  a?  +  a;  =  12,  5.  x2  +  9x  =  -20. 
x>  +  x  +  $y  =  12  +  (by.  6.  arJ-lla;=-28. 

2.  a?-3a;  =  10,  7.  a?  +  13x  =  -42. 
a?-3a;+(f)2=10+(f)2.  8.  a2 -15  a;  =76. 

3.  a?  +  5a;  =  -4.  9.  a?-17a;  =  18. 

4.  a?-7a;  =  8.  10.  x2  +  19 x  =  -  18. 

614.  When  a?  has  a  coefficient,  divide  both  members  by 
the  coefficient.  3  a?  +  9  a;  =  84. 

Dividing  by  3,  x2  +  Sx  =  28. 

Completing  the  square, 

x*  +  3x  +  (f)2  =  28  +  f  =  112  +  9  =  121t 

4  4 

Extracting  square  root,  x  +  f  =  ±  ty. 

...ar=¥-f  =  *  =  4;  or  ~y -*  =  -¥  =  - *• 

^4ns.  4  or  —  7. 

615.  Written  Exercises. 

1.  6a2-    6a;  =  36.  6.  3a?  +   9a;  =  54. 

2.  9a?  +    9a;  =  180.  7.  8 a2  -  72 x  =  -  160. 

3.  7a?  +  28a;=147.  8.  7a?  +  49a;  =  56. 

4.  4a?-40a;  =  -64.  9.  3 x2  +  21  a;  =  54. 

5.  83? -16a;  =504.  10.  5a? -25a;  =  -20. 

616.  Five  times  nothing  =  ? 
Zero  multiplied  by  one  million  =  ? 

If  a;»=5, 

a;-5  =  ? 

lQ(as  -  5)  =  ? 

(.T  +  5)(a;  _  5)  =  ? 


Algebraic  Equations.  477 

If  one  of  two  factors  is  zero,  the  product  is  zero. 
The  converse  is  also  true. 

If  the  product  of  two  factors  is  zero,  one  of  the  factors  is  zero. 
Given  (x  -  2)(x  -  3)  =  0. 

One  of  the  factors  in  the  above  equation  is  equal  to  zero. 
If  x  -  2  =  0, 

by  transposing  we  get  x  =  2. 

If  x  -  3  =  0, 

a;  =  3. 

617.  A  quadratic  equation  may  sometimes  be  readily 
solved  by  factoring. 

1.       x>-5x        =  -6.  2.  x*-  5x  =  14. 

^-5^  +  6  =  0.  x2-5x-U  =  0. 

(x  -  S)(x  -  2)  =  0.  (a;  -  7)(x  +  2)  =  0. 

x  =  3  or  2.  x  =  7  or  —  2. 

Solve  by  factoring: 

3.  a2  +  a- 6  =  0.  8.  z2  -  4z  +  7  =  19. 

4.  a?  +  2x-3  =  0.  9.  ?/2  + 10  =  28  +  3y. 

5.  3^-3^  +  2  =  12.  10.  ^-2^-24  =  0. 

6.  y2  +  7y  +  15  =  3.  11.  ^-15^  =  16. 

7.  ar>- 7  a; +20  =  8.  12.  2/2  +  19?/  =  20. 

618.  Written  Problems. 

1.   The  sum  of  two  numbers  is  12;  their  product  is  32. 
What  are  the  numbers  ? 

x  and  12  —  x  =  numbers.     (12  —  x)  x  =  product. 


2.   The  base  of  a  rectangle  is 
60  feet  longer  than  its  altitude,  x 
Its  area  is  2400  square  feet.    How 
long  is  the  base  ? 


Area  x  2  +  50  X 
2400  sq.  ft. 


x+  50 


478 


Chapter  Seven. 


3.  The  perpendicular  of  a  right-angled  tri- 
angle measures  15  yards  more  than  the  base. 
The  hypotenuse  is  75  yards.  Find  the  length 
of  the  perpendicular. 

x2  +  (15  +  xy  =  752. 

4.  The  hypotenuse  of  a  right-angled  tri- 
angle is  1J  times  as  long  as  the  base.  The 
area  of  the  triangle  is  150  square  yards.  How- 
long  is  the  hypotenuse  ? 

Perpendicular  =  V(£x)2-  x2  ;  area  =  £  base  x  per- 
pendicular. 

5.  The  entire  surface  of  a  square  prism  is  170  square 
feet.  Its  altitude  is  6  feet,  and  one  side  of  its  base  is  x 
feet.     Find  the  value  of  x. 


50+2Z 


6.  A  garden  50  feet  long,  40  feet 
wide,  has  a  walk  just  outside  it  x  feet 
wide.     Find  the  area  of  the  walk. 

If  the  area  of  the  walk  is  784  square 
feet,  what  is  its  width  ? 


7.  A  field,  ABCD,  contains 
12  acres.  Its  length  is  1J  times 
its  breadth.  How  many  rods 
long  is  the  diagonal  BG  ? 


8.  A  flag-staff,  AB,  50  feet  high,  was 
broken  off  at  the  point  C.  The  broken 
part,  resting  on  G,  reached  the  ground 
D,  30  feet  from  the  base  of  the  staff. 
Find  the  length  of  the  part  broken  off. 


Algebraic  Equations. 


479 


9.  A  ladder,  CE  or  DE,  placed  at 
a  point  E,  in  a  street  58  feet  wide  be- 
tween the  opposite  houses,  just  touches 
the  top  of  a  house,  DB,  60  feet  high  on 
one  side  of  the  street,  or  the  top  of  a 
house,  CA,  56  feet  high  on  the  other 
side.     Find  the  length  of  the  ladder. 

DE2  =  602+  (58  -  x)'2=  CE2  =  562  +  a?. 


10.  ABC  is  a  triangle.  The  side 
AB  measures  13  feet ;  the  side  BC,  4 
feet ;  AC,  15  feet.  Find  the  altitude 
BD. 


BD* 


AD2  =  BC2-CD2. 


DxQ 


11.  ABCD  is  a  trapezium.  AB  = 
34  feet ;  BC  =  20  feet ;  CD  =  40  feet ; 
DA  =  26  feet.  The  perpendicular  BF 
measures  16  feet.  Find  the  length  of 
the  diagonal  AC  and  of  the  perpen- 
dicular ED. 


CHAPTER  VIII. 

GEOMETBY. 

619.  Vertical  Lines. 

Hang  a  weight  from  a  fixed  point  by  a  string.  When  the 
weight  stops  swinging  the  string  is  in  a  vertical  line.  What 
way  does  the  lower  end  of  the  string  point  ?  the  higher  end  ? 
Hold  a  sheet  of  ruled  paper  so  that  the  lines  are  vertical. 

620.  Oblique  and  Horizontal  Lines. 

Hold  a  pointer  so  that  it  points  upward  but  not  straight 
up.     It  is  in  an  oblique  line. 

Hold  a  pointer  so  that  it  does  not  point  or  slant  either  up 
or  down.     It  is  in  a  horizontal  line. 

Note.  — In  representing  vertical,  horizontal,  or  oblique  lines  on  the 
page  of  a  book  or  a  sheet  of  paper  it  is  assumed  that  the  book  or  paper 
is  held  in  an  upright  position. 

621.  Oral  Exercises. 

1.  What  kind  of  line  is  represented  by  the  course  of  a 
drop  of  water  running  down  a  roof  ? 

2.  By  the  course  of  a  falling  raindrop  when  there  is  no 
wind? 

3.  By  the  course  of  a  falling  raindrop  when  there  is  a 
wind? 

4.  By  straws  floating  on  the  surface  of  still  water  ? 
Use  the  object  for  the  four  following  exercises. 

6.  How  many  lines  are  there  in  the  edges  of  a  cube  or 
a  rectangular  box  ? 

480 


Exercises  in  Geometry.  481 

6.  When  the  cube  is  placed  on  a  level  table,  how  many- 
edges  are  vertical  ?  How  many  are  horizontal  ?  How  many 
are  oblique  ? 

7.  Hold  the  cube  so  that  four  edges  are  horizontal. 
How  many  are  vertical  ?     How  many  are  oblique  ? 

8.  Hold  the  cube  so  that  no  edges  are  horizontal.  How 
many  are  oblique  ?     How  many  are  vertical  ? 

9.  A  straight  line  is  3  feet  long.  What  kind  of  line  is 
it  if  one  end  is  4  feet  from  the  floor  and  the  other  end  is 
1  foot  from  the  floor  ? 

10.  If  one  end  of  a  3-foot  straight  line  is  4  feet  from  the 
floor  and  the  other  end  is  2  feet  from  the  floor,  what  kind  of 
line  is  it  ? 

11.  If  each  end  of  a  straight  line  is  5  feet  from  the  floor, 
what  kind  of  line  is  it  ? 

12.  If  one  end  of  a  straight  line  is  4  feet  from  the  floor 
and  the  middle  is  4  feet  from  the  floor,  what  kind  of  line 
is  it? 

13.-  A  vertical  straight  line  is  5  feet  long.  The  middle  is 
5  feet  from  the  floor.      How  far  is  each  end  from  the  floor  ? 

14.  A  vertical  line  is  4  feet  long.  One  end  is  5  feet  from 
the  floor.  How  far  from  the  floor  is  the  other  end  ?  Why 
are  there  two  answers  ? 

622.   Angles. 

When  the  ends  of  two  straight  lines  meet  they  form  an 
angle.  A\ 

What  two  lines  form  the  angle  ABC  in 
the  above  figure  ? 

At  what  point  do  they  meet  ?  *2? 

The  point  B  is  the  vertex  of  the  angle  ABC. 

What  is  the  vertex  of  an  angle  ? 


482  Chapter  Eight. 

623.  Designation  of  Angles. 

The  angle  formed  by  the  lines  ST  arid.  TU  may  be  called 
the  angle  T.     It  is  frequently  better  to 
call  it  the  angle  STU  or  UTS,  the  letter 
at  the  vertex  being  placed  between  the 
two  others. 

The  use  of  the  three  letters  is  necessary 
where  two  or  more  angles  have  vertices  /^ 

at  the  same  point,  as  in  the  accompanying  /    ^^ 

figure,  where  UX,  VX,  and  WX  meet  at      x^^__ _ w 
the  point  X. 

624.  Exercises. 

Draw  a  horizontal  line  3  inches  long.  Mark  a  point  in 
this  line  one  inch  from  the  left  end.  From  this  point  draw 
a  line  upward  slanting  towards  the  right.  Mark  each  end 
of  each  line  by  a  letter.  How  many  angles  have  you  formed  ? 
Designate  each  of  these  angles  by  three  letters. 

Note.  —  The  above  exercise  may  be  varied  for  blackboard  drill  — 
draw  a  vertical  line  11  inches  long  ;  mark  a  point  4  inches  from  the 
top ;  draw  a  line  to  the  left  slanting  downward,  etc. 

How  many  angles  are  formed  when  two  lines  meet  at 
their  ends  ?  When  two  lines  pass  through  the  same  point  ? 
When  from  a  point  in  one  line  another  line  is  drawn  ? 

625.  Circular  Measure. 

60  seconds  (")       1  minute. 
60  minutes  (')        1  degree. 
360  degrees  (°)        1  circle,  or  circumference. 

626.  Exercises. 

1.  What  part  of  a  circumference  is  180°?  90°?  60°? 
30°?    45°?    36°?    72°? 

2.  1°  on  the  circumference  of  a  circle  is  5  inches.  What 
is  the  length  of  the  circumference  ? 


Exercises  in  Geometry. 


483 


3.  The  circumference  of  a  circle  is  9000  feet.  1°  =  ? 
1'=? 

4.  How  many  degrees  are  there  between  the  XII  and  the 
I  on  the  face  of  a  clock  ?  between  the  XII  and  VI  ?  between 
the  XII  and  III  ?  between  the  III  and  VII  ? 

5.  If  one  degree  of  the  earth's  circumference  is  69^-  miles, 
find  the  circumference. 

6.  Through  how  many  degrees  does  the  minute  hand  of  a 
clock  pass  in  1  hour?  in  \  hour?  in  15  minutes?  in  5 
minutes  ?  in  10  minutes  ?  in  1  minute  ?  in  3  minutes  ? 


627.   Angular  Measure. 

The  angle  at  the  centre  of  a  circle  has  the  same  number 
of  degrees  as  the  arc  between  the  sides  of  the  angle. 

Thus,  in  the  following  figure  the  angle  AOC  has  the  same 
number  of  degrees  as  the  arc  ABC 


The  circumference  of  this  circle  is  divided  into  36  equal 
parts.  How  many  degrees  are  there  in  each  part  ?  How 
many  degrees  are  there  in  each  of  the  following  angles  ? 

AOB,  BOC,  COD,  DOE,  EOF,  FOG,  GO A,  AOE,  DOF. 


484  Chapter  Eight. 

628.   The  Protractor. 

The  number  of  degrees  in  an  angle  may  be  measured  by  a 
protractor. 

D 

unnnm 


SEMICIRCULAR   PROTRACTOR 


To  measure  an  angle,  XYZ,  for  instance,  produce  the 
lines  YX  and  YZ.  Place  the  point  A  of 
the  protractor  on  the  vertex  (Y)  of  the 
angle,  and  the  edge  AC  on  the  line  YZ 
produced.  Using  the  lower  line  of  figures, 
read  off  from  the  protractor  the  number  of 
degrees  at  the  point  where  the  line  YX  produced  cuts  the 
semicircle. 

In  measuring  the  angle  DEF,  the  line  AB     D* 
is  placed  on  EF,  the  point  A  on  the  vertex 

E.     The  number  of  degrees  in  this  case  is     p ^ 

read  from  the  upper  row  of  figures. 

Note.  — There  is  only  one  point  on  the  protractor  where  the  num- 
bers of  the  upper  and  lower  lines  of  figures  are  equal.  What  is  the 
number  of  degrees  at  that  point  ?  What  kind  of  angle  is  measured 
at  that  point  ?  If  an  angle  is  acute,  would  you  read  its  measure  by 
the  larger  or  by  the  smaller  number  ? 


Exercises  in  Geometry.  485 

EXERCISES  IN  CONSTRUCTION. 

629.  Note.  — In  the  following  exercises,  the  ruler,  the  compasses, 
and  the  protractor  may  be  used. 

The  drawing  should  be  carefully  done  with  a  sharp,  hard  pencil. 

1.  Draw  an  obtuse  angle  formed  by  two  lines,  each  one 
inch  long.     Draw  an  acute  angle  formed 
by  two   lines,   each    six   inches   long. 
Which  is  the  larger  ? 

2.  The  lines  OH  and  IJ  intersect  at 
K,  making  four  right  angles.  Which 
arc  is  longer,  7  8  or  cd  ?  Which  con- 
tains the  greater  number  of  degrees? 

3.  Draw  two  lines  meeting   at  an 
angle  of  45°.     Two  lines  meeting  at  an  angle  of  90°.     Two 
meeting  at  an  angle  of  135°. 

4.  Draw  two  lines  making  two  angles,  one 
of  which  measures  60°.  How  many  degrees 
does  the  other  angle  contain  ? 

5.  To  a  horizontal  line  draw  a  line  making  two  equal  ad- 
jacent angles.     How  many  degrees  does  each  angle  contain  ? 

Two  angles  are  said  to  be  adjacent  when  they  have  one  side  in 
common. 

To  a  vertical  line  draw  a  line  making  two  equal  adjacent 
angles.     How  many  degrees  does  each  angle  contain  ? 

To  an  oblique  line  draw  a  line  making  two  equal  adjacent 
angles.     How  many  degrees  does  each  angle  contain  ? 

6.  How  many  degrees  are  there  in  a  right  angle  ? 

7.  To  an  oblique  line  draw  a  line  making  two  unequal 
adjacent  angles.  How  many  degrees  are  there  in  the  sum 
of  the  two  angles  ? 

Two  angles  are  said  to  be  supplementary  when  they  are  together 
equal  to  two  right  angles. 


486 


Chapter  Eight. 


•fc 


8.  How  many  degrees  in  the  angle  Ty  if 
8  contains  75°  ? 

V  measures  110°.  How  many  degrees  does 
U  measure  ? 

If  one  of  two  supplementary  angles  meas- 
ures 63£°,  how  many  degrees  are  there  in  the 
other  angle  ? 

How  many  degrees  are  there  in  an  angle  supplementary 
to  one  of  47°  45'? 

9.  Construct  angle  5,  60°;  angle  4,  50°. 
Measure  angle  3. 

How  many  degrees  and  minutes  will  there 
be  in  angle  5  when  3  contains  49£°  and  4  contains  83J°  ? 

When  angle  3  contains  36°  30'  and  angle  5  contains 
79°  45',  how  many  degrees  and  minutes  will  angle  4  contain  ? 

10.  Erect  a  perpendicular  at  each  extremity  of  a  hori- 
zontal line.  At  each  extremity  of  a  vertical  line.  At  each 
extremity  of  an  oblique  line. 

Note.  —  A  line  making  a  right  angle  with  another  line  is  said  to  be 
perpendicular  to  it. 

11.  Construct  a  square  upon  a  horizontal  line, 
oblique  line. 

12.  Draw  two  lines  intersecting  at  an 
angle  of  100°.  Mark  in  each  of  the  other 
three  angles  the  number  of  degrees  it  con- 
tains. 

13.  Draw  two  lines  making  an  angle  (6) 
of  150°.  Construct  an  adjacent  angle  (7) 
containing  80°.  How  many  degrees  will 
angle  8  contain? 

14.  How  many  degrees  will  there  be  in  the 
sum  of  five  angles  having  the  same  vertex  ? 


Upon  an 


Exercises  in  Geometry.  487 

15.  Draw  five  equal  angles  having  a  common  vertex. 

16.  Draw  six  equal  angles  having  a  common  vertex.  Is 
any  angle  supplementary  to  the  angle  next  it  ?     Why  ? 

Are  any  of  the  angles  vertical  ?     Why  ? 

17.  Draw  two  angles,,  one  of  65°  and  the  other  of  25°. 
Draw  a  third  angle  equal  to  the  sum  of  both. 

Draw  an  angle  equal  to  their  difference. 

18.  Draw  an  angle  equal  to  the  sum  of  three  angles 
measuring,  respectively,  40°,  50°,  and  60°. 

630.   Parallels. 

Lines  which  lie  in  the  same  plane  and  which  cannot  meet, 
no  matter  how  far  produced,  are  said  to  be  parallel. 

19.  Using  the  protractor,  draw  two  or  more  lines  that 
shall  be  perpendicular  to  a  horizontal  line.  Where  will  they 
meet  ? 

Draw  two  or  more  that  shall  be  perpendicular  to  a  verti- 
cal line.     Where  will  they  meet  ? 

Draw  two  or  more  that  shall  be  perpendicular  to  an 
oblique  line.     Where  will  they  meet  ? 

20.  To  a  horizontal  line  draw  two  or  more  lines  running 
in  the  same  direction,  and  each  making  an  angle  of  35°  with 
the  first  line.     Will  the  oblique  lines  meet  ? 

Draw  two  or  more  lines  running  in. the  same  direction, 
and  each  making  an  angle  of  125°  with  a  vertical  line.  Will 
the  oblique  lines  meet  if  produced  very  far  ? 

Draw  two  or  more  lines  running  in  the  same  direction, 
and  each  making  an  angle  of  74°  with  an  oblique  line.  Will 
the  former  lines  meet  ? 

21.  Draw  two  lines  making  angles  of  30°  and  60°,  respec- 
tively, with  a  third  line.  Will  the  two  former  lines  meet  if 
produced  in  either  direction  ? 


488 


Chapter  Eight. 


22.  Draw  a  line,  AB,  meeting  a  horizontal  line,  BO,  at 
an  angle  of  58°.     Draw  a  third  line,  DE, 
parallel  to  the  horizontal  line,  and  cut- 
ting the  oblique  line.    What  angles  does 
it  make  with  the  oblique  line  ? 

Draw  a  fourth  line,  EG,  parallel  to 
the  oblique  line,  and  cutting  both  hori- 
zontal lines. 

Mark  in  each  of  the  twelve  angles  the  number  of  degrees 
it  contains. 

23.  QR  and  UV  are  parallel  lines, 
cut  by  a  line  ST.  If  the  angle  b 
measures  50°,  how  many  degrees  does 
a  measure  ? 

Find  the  number  of  degrees  in  each 
of  the  other  six  angles. 

631.   Triangles. 

24.  From  the  extremities  of  the  line  AB,  draw  lines  that 
shall  make  angles  of  60°  and  40°,  respec- 
tively, with  AB.     Prolong  the  lines  until 
they  meet  at  C,  forming  a  triangle. 

Measure  the  angle  at  C.     How  many 
degrees   does   it   contain?      How    many 
degrees  are  there  in  the  sum  of  the  three  angles  of  the  tri- 
angle ? 

25.  Construct  a  triangle  having  one  angle  of  90°  and  one 
of  30°.     Measure  the  third  angle. 

How  many  degrees  are  there  in  the  sum 
of  the  three  angles  ? 

26.  Construct  a  triangle,  KLM, 
making  the  angles  at  the  base  28° 
and  120°,  respectively.  Draw 
NO,  parallel  to  LM. 


Exercises  in  Geometry.  489 

Is  the  angle  e  equal  to  any  angle  of  the  triangle  ?  How 
many  degrees  does  it  contain  ?  Is  the  angle  /  equal  to  any 
angle  of  the  triangle  ?     How  many  degrees  does  it  contain  ? 

How  many  degrees  are  there  in  the  sum  of  the  angles  e,  g, 
and  /?     How  many  degrees  are  there  in  the  angle  g  ? 

27.  How  many  degrees  are  there  in  the  three  angles  of 
any  triangle  ? 

28.  Two  angles  of  a  triangle  measure  36°  and  65°,  respec- 
tively.    How  many  degrees  does  the  third  angle  contain  ? 

29.  Draw  a  triangle  containing  two  angles  of  50°  and  70°, 
respectively.  How  many  degrees  are  there  in  the  third 
angle  ? 

Measure  each  side,  and  mark  on  the  side  its  length. 
Opposite  which  angle  is  found  the  longest  side  ?     Opposite 
which,  the  shortest  side  ? 

30.  Draw  a  triangle  having  two  angles  of  75°  each.  Are 
any  two  of  its  sides  equal  ? 

Draw  a  triangle  having  two  angles  of  50°  each.  Are  any 
of  its  sides  equal  ? 

31.  Draw  a  triangle  having  two  angles  of  60°  each.  How 
many  degrees  does  the  third  angle  contain  ? 

Are  any  of  its  sides  equal  ? 

32.  If  a  triangle  has  two  of  its  sides  equal,  what  is  true 
of  its  angles? 

33.  If  a  triangle  has  three  of  its  sides  equal,  what  is  true 
of  its  angles? 

632.  A  triangle  having  all  its  sides  equal,  is  called  an 
equilateral  triangle. 

A  triangle  having  two  equal  sides,  is  called  an  isosceles 
triangle. 

A  triangle  having  all  its  sides  unequal,  is  called  a  scalene 
triangle. 


49° 


Chapter  Eight. 


34.  How  does  a  perpendicular  let  fall  upon  the 
of  an  isosceles  triangle  from  the  opposite 
angle  divide  the  angle  ?  How  does  it  divide 
the  base  ?  How  do  the  angles  at  the  base 
of  an  isosceles  triangle  compare  with  each 
other  as  to  size  ? 

The  unequal  side  of  an  isosceles  triangle  is  called 
the  base. 

35.  Draw  an  isosceles  triangle  having  the  base  a  vertical 
line. 

An  isosceles  triangle  having  the  vertex  below  the  base. 
One  having  an  oblique  line  for  the  base. 

36.  Draw  a  right-angled  isosceles  triangle.     How  many- 
degrees  will  there  be  in  each  of  the  other  angles  ? 

Draw  an  obtuse-angled  isosceles  triangle. 

37.  How  many  degrees  will  there  be  in  each  angle  of  an 
equilateral  triangle  ? 

Draw  an  equilateral  triangle  having  one  side  vertical. 
Draw  an  equilateral  triangle  having   jy  q  ^ 

its  vertex  below  the  base. 

38.  DEF  is  an  isosceles  triangle, 
DF  and  EF  being  the  equal  sides.  If 
the  angle  1  measures  50°,  how  many 
degrees  are  there  in  each  of  the  other 
five  angles,  when  the  line  FO  bisects 
the  base.? 

39.  ABC  is  a  right-angled  A 
triangle,  the  angle  at  B  measur- 
ing 90°,  and  the  angle  at  C 
measuring  30°.  If  the  line  AX 
is  so  drawn  as  to  make  the 
angle  AXB  equal  to  60°,  find 
the  number  of  degrees  in  the 
angles  m,  n,  and  p,  respectively. 


Exercises  in  Geometry.  491 

633.   Quadrilaterals. 

A  plane  figure  of  four  sides  is  called  a  quadrilateral. 

When  the  opposite  sides  are  parallel,  the  quadrilateral  is 
called  a  parallelogram.     (Figs.  1  to  8.) 

A  rectangle  is  a  parallelogram  all  of  whose  angles  are  right 
angles.     (Figs.  1  to  4.) 

When  the  four  sides  of  a  rectangle  are  equal  to  each  other, 
it  is  called  a  square.     (Figs.  1  and  2.) 

The  term  oblong  is  frequently  applied  to  rectangles  whose 
adjacent  sides  are  unequal.     (Figs.  3  and  4.) 


Fig.  1. 


Fig.  2. 


Fig.  3. 


Fig.  4. 


A  rhombus  is  a  parallelogram  all  of  whose  sides  are  equal, 
but  whose  angles  are  oblique.     (Figs.  5  and  6.) 

When  the  adjacent  sides  of  a  parallelogram  are  unequal 
and  the  angles  are  oblique,  it  is  called  a  rhomboid.  (Figs.  7 
and  8.) 


VA 


Fig.  5. 


Fig.  6. 


Fig.  7. 


Fig.  8. 


A  trapezoid  is  a  quadrilateral  having  only  two  of  its  sides 
parallel.     (Figs.  9  and  10.) 

A  trapezium  is  a  quadrilateral  having  no  two  sides  paral- 
lel.    (Figs.  11  and  12.) 


Fig.  9. 


Fig.  10. 


Fig.  11. 


Fig.  12. 


492  Chapter  Eight. 

634.   The  altitude  of  a  parallelogram  is  the  perpendicular 
distance  between  its  base  and  the  side  opposite. 


M 


G 


x  j;       b-       IT 


The  altitude  of  a  triangle  is  the  perpendicular  distance 
between  the  vertex  and  the  base,  or  between  the  vertex  and 
base  produced. 

AB  is  the  altitude  of  MANT;  DX  is  the  altitude  of 
DBE-,  OYoi  OHI. 

40.  Draw  a  parallelogram.  How  many  angles  does  it  con- 
tain? Into  how  few  triangles  can  you  divide  a  parallelo- 
gram? How  many  degrees  are  there  in  the  sum  of  the 
angles  of  each  triangle  ?  How  many  degrees  are  there  in 
the  sum  of  the  angles  of  a  parallelogram  ? 

41.  Construct  a  parallelogram,  the  adjacent  sides  of  which 
shall  measure  2  inches  and  3  inches,  respectively,  and  the 
angle  between  them  60°.  How  long  will  each  of  the  other 
two  sides  be?  Measure  each  of  the  other  angles.  How 
many  degrees  are  there  in  the  sum  of  the  four  angles  ? 

42.  Construct  a  trapezoid  having  a  base  of  5  inches,  alti- 
tude 3  inches,  the  angles  at  the  base  measuring  90°  and  60°, 
respectively.  Measure  the  remaining  angles,  and  find  the 
sum  of  the  four  angles.  How  long  is  each  of  the  remaining 
sides  ? 

43.  Fold  a  piece  of  paper  twice  at  right  angles,  and  cut 
off  the  folded  corner,  making  a  rhombus  when  the  part  cut 
off  is  opened  out. 

Can  you  cut  out  a  rhombus  having  two  angles  of  60°  each  ? 
A  rhombus  having  two  angles  of  80°  each  ? 


Exercises  in  Geometry.  493 

44.  Can  you  so  cut  a  piece  of  paper,  folded  twice  at  right 
angles,  that  the  part  cut  off  will  be  a  square  ? 

45.  Draw  a  rectangle,  base  2\  inches,  altitude  2  inches. 
A  rhomboid,  base  2\  inches,  altitude  2  inches. 

46.  Make,  out  of  paper,  a  rectangle  and  a  rhomboid,  each 
having  the  above  dimensions,  and  endeavor  to  ascertain,  by 
cutting,  whether  or  not  they  are  equal  to  each  other  in  area. 

635.  The  Circle. 

47.  Draw  a  circle.  Between  two  points  on  the  circum- 
ference draw  a  line  that  does  not  pass  through  the  centre. 

This  line  is  called  a  chord. 

48.  Draw  a  circle.  In  it  draw  .two  diameters,  a  radius, 
and  three  chords.     Write  on  each  line  its  name. 

49.  Draw  a  part  of  the  circumference  of  a  circle  greater 
than  one-half  of  it.     Draw  the  chord. 

A  part  of  the  circumference  is  called  an  arc. 

50.  Draw  an  arc  less  than  a  semi-circumferenee.  Draw  a 
chord.     Write  the  name  on  each. 

Can  you  make  a  chord  that  will  be  longer  than  the 
diameter  ? 

51.  Draw  two  equal  circles.  In  the  first  draw  the  chord 
of  an  arc  of  120°.  In  the  second,  the  chord  of  an  arc  of 
240°.  What  is  the  ratio  between  the  two  chords  you  have 
drawn  ? 

52.  In  a  circle  draw  a  chord  equal  in  length  to  the  radius. 
How  many  degrees  are  there  in  the  arc  whose  chord  has 
been  drawn  ? 

53.  Draw  an  arc  of  72°.  To  its  extremities  draw  two 
radii. 

The  part  of  the  surface  of  a  circle  enclosed  by  two  radii  and  the 
intercepted  arc  is  called  a  sector. 


494  *       Chapter  Eight. 

54.  Draw  a  sector  of  60°  (a  sextant).  A  sector  of  90°  (a 
quadrant). 

55.  Draw  an  arc  of  120°.     Draw  the  chord. 

The  part  of  the  surface  of  a  circle  bounded  by  an  arc  and 
its  chord  is  called  a  segment. 

56.  Draw  several  circles  having  the  same  centre,  but  of 
unequal  radii  (concentric  circles). 

57.  Draw  two  equal  circles  just  touching  each  other  (tan- 
gent).    Draw  two  unequal  circles  tangent  to  each  other. 

Within  a  large  circle  draw  a  smaller  one  tangent  to  it. 

58.  Draw  circles  of  equal  radii  cutting  each  other.  Draw 
intersecting  circles  of  unequal  radii. 

636.  Pentagons,  Hexagons,  Octagons. 

59.  Divide  the  circumference  of  a  circle  into  four  equal 
arcs.     Draw  the  chords,  forming  an  inscribed  square. 

60.  If  you  wish  to  inscribe  in  a  circle  a 
figure  of  five  equal  sides,  into  how  many  equal 
arcs  must  the  circumference  be  divided  ?  How 
many  degrees  will  each  arc  contain  ? 

637.  A  plane  figure  bounded  by  straight  lines  is  called  a  polygon. 

A  five-sided  polygon  is  called  a  pentagon  ;  one  of  six  sides,  a  hexa- 
gon; of  seven,  a  heptagon;  of  eight,  an  octagon;  of  nine,  a  nonagon; 
of  ten,  a  decagon;  etc. 

A  regular  polygon  is  one  that  is  both  equilateral  and  equi- 
angular. 

61.  Inscribe  a  regular  pentagon  in  a  circle.  Use  the 
protractor. 

62.  Inscribe  in  a  circle  a  regular  hexagon.  A  regular 
octagon.     An  equilateral  triangle. 


Exercises  in  Geometry.  495 

63.  Inscribe  in  a  circle  a  regular  hexagon.     Connect  the 
opposite  corners  by  lines  passing  through  the 
centre  of  the  circle,  forming  six  triangles. 

How  many  degrees  are  there  in  each  of  the 
six  angles  about  the  centre  of  the  circle  ?  In 
each  of  the  twelve  angles  at  the  circumference  ? 
How  many  degrees  are  there  in  the  sum  of  angles  1  and  2  ? 

Is  each  of  the  six  triangles  scalene,  equilateral,  or 
isosceles  ? 

64.  Divide  a  regular  inscribed  pentagon  into  five  equal 
triangles  by  lines  drawn  from  the  centre  of  the  circle. 

What  kind  of  triangles  are  formed;  isosceles,  scalene,  or 
equilateral  ? 

How  many  degrees  are  there  in  each  angle  at  the  centre  ? 
In  each  angle  at  the  circumference?  How  many  degrees 
are  there  in  the  sum  of  two  adjoining  angles  at  the  circum- 
ference ?     In  each  angle  of  the  pentagon  ? 

65.  About  a  circle  circumscribe  a  square.  An  equilateral 
triangle.  A  regular  pentagon.  A  regular  hexagon.  A 
regular  octagon. 

PROBLEMS  IN  CONSTRUCTION. 

638.  In  drawing  the  following  exercises,  only  the  ruler  and  the 
compasses  are  to  be  used.    Use  neither  the  protractor  nor  the  triangle. 

66.  Draw  a  circle,  radius  an  inch  and  a  half.  Outside  of 
it,  and  tangent  to  it,  draw  a  second  circle  of  an  inch  radius. 

How  far  apart  are  the  centres  ? 

67.  Draw  two  tangent  circles  having  radii  of  an  inch  and 
a  half  and  an  inch,  respectively,  one  within  the  other. 

How  long  is  the  line  joining  the  centres  ? 

68.  With  centres  3  inches  apart  draw  two  equal  circles 
tangent  to  each  other.     How  long  is  the  radius  of  each  ? 


49^  Chapter  Eight. 

69.  With  centres  three  inches  apart  draw  two  equal 
circles  of  2  inches'  radius.     Connect  the  centres. 

Draw  a  line  joining  the  two  points  in  which  the  circles 
intersect.  How  does  this  line  divide  the  line  connecting 
the  centres  ? 

Draw  radii  from  each  centre  to  each  point  of  intersection. 

70.  Construct  an  isosceles  triangle,  base  3  inches,  equal 
sides  2  inches. 

Note.  —  Use  circles  or  arcs  where  necessary. 

71.  Construct  an  isosceles  triangle,  base  3£  inches,  equal 
sides  4  inches. 

Divide  it  into  two  equal  parts.  Do  not  locate  the  centre 
of  the  base  by  measurements. 

72.  On  a  vertical  line  construct  an  isosceles  triangle. 
Without  measuring  the  length  of  the  base  draw  a  perpen- 
dicular to  the  centre  of  the  base. 

73.  Bisect  a  vertical  line.     An  oblique  line. 
Do  not  measure  the  length  of  the  line. 

74.  Construct  an  equilateral  triangle  on  a  two-inch  line. 

75.  Construct  an  equilateral  triangle  on  a  vertical  line. 
On  an  oblique  line. 

76.  Cut  out  two  equal  right-angled  triangles.  Put  them 
together  in  different  ways  so  as  to  form  two  different 
isosceles  triangles. 

0 

77.  Construct  a  scalene  triangle. 

A  triangle  having  sides  measuring  1,  1J,  2  inches,  respec- 
tively. 

One  whose  sides  measure  2,  2|,  and  3  inches,  respectively. 

78.  Can  you  construct  an  isosceles  triangle  whose  base 
measures  4  inches,  equal  sides  2  inches  ? 

Try  to  construct  a  scalene  triangle  with  sides  measuring 
1,  2,  and  3  inches,  respectively. 


Exercises  in  Geometry.  497 

79.  Draw  a  circle.     In  it  draw  a  chord. 

Bisect  the  chord,  using  as  few  lines  and  as  short  ones  as 
you  can. 

Note.  —  Do  not  use  the  ruler  to  ascertain  the  length  of  the  chord 
before  bisecting  it. 

80.  Divide  a  sector  into  two  equal  parts. 

81.  Draw  a  circle.  Draw  a  chord.  Draw  a  radius  through 
the  centre  of  the  chord. 

Is  the  radius  perpendicular  to  the  chord  ?     Why  ? 

82.  Bisect  the  arc  of  a  circle  and  its  chord. 
Bisect  the  arc  of  a  circle  without  drawing  the  chord. 

83.  Draw  a  perpendicular  to  the  middle  point  of  a  hori- 
zontal line.  To  the  middle  point  of  a  vertical  line.  To  the 
middle  point  of  an  oblique  line. 

84.  Draw  in  a  circle  two  diameters  perpendicular  to  each 
other. 

85.  Divide  the  circumference  of  a  circle  into  four  equal 
parts.     Into  eight  equal  parts. 

Inscribe  a  square  in  a  circle. 

86.  Inscribe  a  regular  octagon  in  a  circle. 

87.  Connect  the  opposite  vertices  of  a  regular  octagon 
inscribed  in  a  circle  by  lines  passing  through  the  centre  of 
the  circle. 

Lines  connecting  the  opposite  vertices  of  a  polygon  are  called 
diagonals. 

88.  Inscribe  a  square  in  a  circle.  Circumscribe  a  square 
whose  sides  shall  be  perpendicular  to  the  diagonals  of  the 
inscribed  square. 

89.  Cut  out  the  circumscribed  square  and  show  by  folding 
that  it  is  twice  the  area  of  the  inscribed  square. 


498  Chapter  Eight. 

90.  Construct  an  equilateral  triangle  on  a  horizontal  line 
1  inch  long.  On  the  right  side  as  a  base,  construct  a  second 
equilateral  triangle.  On  the  left  side  of  the  first  triangle, 
construct  a  third  Construct  three  more,  completing  the 
hexagon. 

91.  Can  you  circumscribe  a  circle  about  the  above  hexa- 
gon ?    What  is  the  radius  of  the  circle  ? 

92.  Inscribe  a  regular  hexagon  in  a  circle  whose  radius  is 
1  \  inch.     What  is  the  length  of  each  side  of  the  hexagon  ? 

93.  Inscribe  in  a  circle  an  equilateral  triangle.  On  each 
of  its  three  sides  construct  an  equilateral  triangle. 

94.  Construct  an  arc  of  60°.  Draw  two  lines  meeting  at 
an  angle  of  60°. 

95.  Bisect  an  arc  of  60°.  Draw  two  lines  meeting  at  an 
angle  of  30°. 

96.  Construct  an  angle  of  60°  and  an  angle  of  30°.  Draw 
two  lines  making  an  angle  equal  to  the  sum  of  the  two  angles 
first  constructed. 

97.  Erect  a  perpendicular  at  the  end  of  a  horizontal  line. 
At  the  end  of  a  vertical  line.     At  the  end  of  an  oblique  line. 

98.  Construct  an  angle  of  45°.  An  angle  of  22£°.  An 
angle  of  135°.     An  angle  of  15°.     An  angle  of  75°. 

99.  Draw  a  circle,  radius  1  inch.  Draw  a  diameter,  and 
produce  it  an  inch  beyond  the  circumference.  At  the  centre 
of  the  circle  erect  a  perpendicular  to  the  diameter. 

100.  An  inch  from  one  end  of  a  3-inch  line,  erect  a  perpen- 
dicular, using  as  few  and  as  short  lines  as  possible. 

101.  Draw  a  horizontal  line.     Take  a  point  above  the  line 
as  a  center.     Draw  an  arc  that  cuts  the  line  in  two  places. 

102.  Draw  a  line.     From  a  point  above  the  line,  let  fall 
a  perpendicular  to  the  line. 


Exercises  in  Geometry.  499 

EQUAL  TRIANGLES.     EQUIVALENT  TRIANGLES. 

639.  Note.  — The  protractor  and  the  triangle  may  be  used  in  the 
following  exercises. 

1.  Draw  a  rectangle,  base  2\  inches,  altitude  2  inches. 
Draw  a  rhomboid,  base  2£  inches,  altitude  2  inches.  Find 
the  area  of  each. 

2.  With  a  base  2 \  inches,  altitude  2  inches,  draw 

(a)  A  right-angled  triangle. 

(b)  An  isosceles  triangle. 

(c)  One  or  more  acute-angled  scalene  triangles. 

(d)  One  or  more  obtuse-angled  triangles. 
Calculate  the  area  of  each. 

3.  Can  you  show,  by  cutting  from  paper,  that  a  right- 
angled  triangle  having  its  base  and  perpendicular  4  inches 
and  3  inches,  respectively,  has  the  same  surface  as  an  acute- 
angled  triangle  whose  base  and  altitude  are  4  inches  and 
3  inches  respectively,  and  an  obtuse-angled  triangle  whose 
base  and  altitude  are  4  inches  and  3  inches,  respectively  ? 

Two  triangles  that  have  the  same  area  are  called  equivalent  tri- 
angles ;  those  having  their  corresponding  sides  and  angles  equal,  each 
to  each,  are  called  equal  triangles. 

4.  Construct  a  triangle  whose  sides  measure  1J,  2,  and 
2\  inches,  respectively.  Construct  another  triangle  having 
its  sides  of  the  same  lengths.  Are  the  angles  of  the  second 
equal  to  the  angles  of  the  first  ?     Are  the  triangles  equal  ? 

5.  Draw  two  triangles  each  of  which  has  two  sides 
measuring  1^  and  3  inches,  respectively,  and  the  included 
angle  60  degrees.  Is  the  third  side  of  one  triangle  equal  to 
the  third  side  of  the  other?  Are  the  remaining  angles 
of  the  first  triangle  equal  to  the  remaining  angles  of  the 
second  ? 


500 


Chapter  Eight. 


6.  Construct  two  triangles  with  equal  bases,  and  angles  at 
the  bases  respectively  equal.     Are  the  triangles  equal  ? 

7.  A  person  wishing  to  ascertain  the  length,  AB,  of  a 
pond,  places  a  pole  at  a  convenient 
point,  G,  visible  from  A  and  B.  The 
distance  BO  is  measured,  and  a  pole  is 
set  up,  on  a  line  with  B  and  G,  at  D, 
the  distance  CD  being  made  equal  to 
BG.  A  pole  is  also  placed  at  E,  on  a 
line  with  A  and  C,  the  distance  CE 
being  made  equal  to  AG. 

Can  you  show  that  the  length,  AB,  of  the  pond  can  be 
ascertained  by  measuring  the  distance  DE  ? 


CALCULATING  HEIGHTS  AND  DISTANCES. 

640.    To  verify  the  results  obtained  by  calculation,  the  pupil  should 
make  diagrams,  drawing  the  figures  to  a  convenient  scale.  (7 

1.   If    AB    in    a    right- 
angled    triangle    measures  ^ 
120  feet,  and  a  perpendicu-                           ^S 
lar,  vw,  erected  10  feet  from 
A  measures  5  feet,  calcu- 
late the  length  of  BG. 

Aw:AB::wv:BC: 


—         w 
i.e.  10  :  120  : 


5:  BG. 


2.    A  post  6  feet  above  ground  throws  a  shadow  of  7£  feet 
How  high  is  a  tree  whose  shadow  measures  60  feet  ? 


t^ 


Exercises  in  Geometry. 


5<» 


3.  Wishing  to  ascertain  the  distance  between  two  houses, 
R  and  S,  on  opposite  sides 
of  a  stream,  I  measure  a 
line,  SV,  at  right  angles  to 
SR,  200  feet.  At  T,  90  feet 
from  V,  the  perpendicular 
TW  measures  60  feet.  Re- 
quired the  distance  SR. 

VT:TW::VS:SR 

4.  Beginning  at  B,  100  feet  from  the  bank  of  a  rive^  a 
line,  BC,  is  measured  1200  feet 
long.  At  D,  distant  from  C 
50  feet,  the  perpendicular  DE 
is  found  to  measure  90  feet. 
What  is  the  distance  from  B  to 
A,  a  tree  on  the  opposite  bank  ? 
How  wide  is  the  river  ? 

5.  A  boy,  whose  eye  (E)  is  4  feet  from  the  ground,  can 
just  see  the  top  (A)  of  a  steeple  when  he  stands  back  3  feet 
from  a  fence  (CG)  6  feet  high.  The  distance  from  the 
foot  of  the  fence  to  the  centre  of  the  base  of  the  steeple  is 
177  feet.     Find  the  height  of  the  steeple,  AB. 

CD  =  ?         EH=?        ED:  CD::  EH:  AH. 

When  AH  is  found,  how  may  you  get  AB  ? 


502 


Chapter  Eight. 


6.  Wishing  to  ascertain  the  distance  AB,  I  measure  a 
line,  AD,  at  right  angles  to  AB, 
12  chains ;  DE,  at  right  angles  to 
AD,  5  chains ;  and  find  that  a  line 
sighted  from  E  to  B  intersects 
AD  at  C,  distant  from  D  3.25 
chains.  What  is  the  distance  from 
AtoB? 

Note.  —The  triangles  DCE  and  ACB 
are  similar.     Why  ? 

7.  Wishing  to  find  the  height  of  a  tower,  //,  I  set  up 
a  pole,  cd,  12  feet  long  above  the  ground.  Another  pole,  ab, 
4J  feet  above  ground,  is  set  up  at  such  a  distance  that  the 
tops  of  the  two  poles 
and  of  the  tower  are 
in  a  line.  The  distance 
between  the  poles  (ae 
or  db)  is  10J  feet.  The 
distance  from  d  to  the 
foot  of  the  tower  is  195 
feet.  The  width  of  the 
tower  Qcj)  is  30  feet. 

The  similar  triangles  aec  and  ahf  give  us  the  proportion 


kij 


ae  :ah  : :  ec:  hf. 

What  is  the  distance  ec?     ah  =  bi=bd+dk+ki 
fh  is  found,  what  must  be  added 
to  get  the  height  of  the  tower  ? 

8.  To  determine  the  height 
of  a  building,  MN,  a  person 
attached  a  straight  strip  of 
wood,  ab,  to  a  post,  OP,  in 
such  a  manner  that  sighting 
from  a,  he  could  just  see  M> 


ki=\kj.     When 


Exercises  in  Geometry. 


503 


the  top  of  the  building.     He  then  sighted  down  from  b,  and 
marked  on  the  ground  the  point  R,  on  a  line  with  ab. 

PQ  was  found  by  measurement  to  be  4  feet,  MP  6  feet, 
PN 120  feet.     Eequired,  MN. 

9.  Wood-choppers,  desiring 
to  know  the  height  of  a  tree 
before  cutting  it,  sometimes 
make  an  isosceles  right-angled 
triangle  of  wood  or  paper,  and 
"  step  off "  the  distance  on  level 
ground  from  the  point  at  which 
they  find  they  can  just  see  the 
top  of  the  tree  looking  along 
the  hypotenuse  of  the  triangle, 
the  base  being  parallel  to  the  ground. 

How  high  is  the  tree  AB,  if  ^4(7  is  36  paces  of 
3  feet  each,  and  the  angle  AGB  is  45°  ? 

10.  B  is  a  point  on  the  bank  of  a  stream 
due  east  of  A  on  the  other  bank.  A  boy 
walks  due  south  of  A  until  he  reaches  a 
point  at  which  he  finds,  from  his  pocket 
compass,  that  he  is  directly  southwest  of 
B.  If  the  distance  AC  measures  119  yards, 
how  wide  is  the  stream  ? 


TABLES 


LINEAR  MEASURE 


12    inches  (in.)  .     . 

3    feet 

5£  yards,  or  16|  feet 
40    rods      .... 
320    rods      .... 


=  1  foot ft. 

—  1  yard yd. 

=  1  rod rd. 

=  1  furlong fur. 

=  1  mile mi. 


1  mi. 


320  rd.  =  1760  yd.  =  5280  ft.  =  63,360  in. 

A  hand,  used  in  measuring  the  height  of  horses,  =  4  in.  A  knot,  used 
in  measuring  distances  at  sea,  =  1.15  mi.  A  fathom,  used  in  measuring 
the  depth  of  the  sea, 


Oft. 


SQUARE  MEASURE 

144  square  inches  (sq.  in.)  .     .  =  1  square  foot  . 

9  square  feet =  1  square  yard  . 

30£  sq.  yd.,  or  272^  sq.  ft.   .     .  =  1  square  rod  . 

160  square  rods =  1  acre      .     .  . 

640  acres      .     .     .     .   <  .     .     .  =  1  square  mile  . 


sq.  ft. 
sq.  yd. 
sq.  rd. 
A. 
sq.  mi. 


1  A.  =  160  sq.  rd.  =  4840  sq.  yd.  =  43,560  sq.  ft. 
A  Section  of  land  is  a  square  mile. 

Koofing,  flooring,  and  slating  are  often  estimated  by  the  square, 
-which  contains  100  square  feet. 

SURVEYORS'  MEASURE 

In  measuring  land,  surveyors  use  a  chain  (ch.)  which  contains  100 
links  (1.)  and  is  4  rods  long.  Since  the  chain  isf4  rods  long,  a  square 
chain  contains  16  sq.  rd.,  and  10  sq.  ch.  =  160  sq.  rd.,  or  1  acre. 


CUBIC  MEASURE 

1728  cubic  inches  (cu.  in.)  .     .  =  1  cubic  foot 

27  cubic  feet =  1  cubic  yard 

128  cubic  feet =  1  cord     .     . 

16  cubic  feet =  1  cord  ft.     . 

8  cord  feet =  1  cord     .     . 


cu.  ft. 
cu.  yd. 
cd. 

cd.  ft. 
cd. 


Note. —  In  computing  the  contents  of  an  enclosing  wall,  masons  and 
brick-layers  regard  it  as  one  straight  wall  whose  length  is  the  distance 
around  it  on  the  outside.    Corners  are  thus  measured  twice. 

A  perch  of  stone  or  masonry  is  16£  ft.  long,  1J  ft.  thick,  and  1  ft. 
high,  and  contains  24 1  cu.  ft. 


YB  35889 


MEASURES  OF   CAPACITY 

Liquid  Measure  Dry  Measure 

4  gills      =  1  pint    .     .     .     pt.         2  pints    =  1  quart    .     .     qt. 

2  pints    sb  1  quart .     .     .     qt.         8  quarts  s=  1  peck     .     .     pk. 

4  quarts  =  1  gallon     .     .    gal.       4  pecks  =  1  bushel .    .     bu. 

The  standard  gallon  contains  231  cubic  inches. 
The  standard  bushel  contains  2150.42  cubic  inches. 

The  capacity  of  cisterns,  reservoirs,  etc.,  is  often  expressed  in  barrels 
(bbl.)  of  315  gallons  each,  or  in  hogsheads  (hhd.)  of  63  gallons  each.  In 
commerce,  these  vary  in  size. 


AVOIRDUPOIS  WEIGHT 


16  ounces  (oz.) 
100  y 
2000] 


1  pound lb. 


The  long 
and  in  weigl 


1  bushel  of 
1  bushel  of  1 
1  bushel  of 
1  bushel  of 


24  gr: 
20  pe 
12  oul 


54884 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


APOTHECARIES'  WEIGHT 

60  grains  (gr. )       .     .     .     .  =  1  dram    .     .     .     dr.,  or  3. 

8  drams =1  ounce  .     .     .     oz.,  or  %. 

12  ounces =  1  pound  .     .     .     lb.,  or  lb. 

One  pound  Apothecaries'  weight  =  576C  grains. 


BRITISH  OR  STERLING  MONEY 

4  farthings =1  penny     .     ....  A 

.     .     .  s. 

.     .     .  £. 


12  pence =  1  shilling 

20  shillings =1  pound 

5  shillings =1  crown. 

The  value  of  £1  is  $4.8665  in  United  States  gold  coin. 

The  unit  of  French  money  is  1  franc,  which  is  19.3  cents.    The  unit  of 
German  money  is  1  mark,  which  is  23.85  cents. 


Ililittil 


1 

1 

■i  11 


i    in! 


1 

Jar 


'111! 


